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Edward Bright 




Mathematics Dept 



THE APPLICATIONS OF ELLIPTIC 

FUNCTIONS. 



THE APPLICATIONS 



OF 



ELLIPTIC FUNCTIONS 



BY 

ALFRED GEORGE GREENHILL 

M.A., F.R.S., PROFESSOR OK MATHEMATICS IN" THE ARTILLERY COLLEGE, WOOLWICH 



MACMILLAN AND CO. 

AND XETT YORK 
1892 

[All riyhlft reserved] 




AY* 









CONTENTS. 



PAGE 

INTRODUCTION, vii 



CHAPTER I. 

THE ELLIPTIC FUNCTIONS, - - - 1 

CHAPTER II. 

THE ELLIPTIC INTEGRALS, - - 30 

CHAPTER III. 

GEOMETRICAL AND MECHANICAL ILLUSTRATIONS OF THE ELLIPTIC 

FUNCTIONS, - 66 

CHAPTER IV. 
THE ADDITION THEOREM FOR ELLIPTIC FUNCTIONS, - - - - 112 

CHAPTER V. 

THE ALGEBRAICAL FORM OF THE ADDITION THEOREM, 142 

CHAPTER VI. 
THE ELLIPTIC INTEGRALS OF THE SECOND AND THIRD KIND, - - 175 

CHAPTER VII. 
THE ELLIPTIC INTEGRALS IN GENERAL AND THEIR APPLICATIONS, - 200 

CHAPTER VIII. 
THE DOUBLE PERIODICITY OF THE ELLIPTIC FUNCTIONS, - - - 254 

781468 



v i CONTENTS. 

CHAPTER IX. 

PAGE 

THE RESOLUTION OF THE ELLIPTIC FUNCTIONS INTO FACTORS AND 

SERIES, ... 277 

CHAPTER X. 
THE TRANSFORMATION OF ELLIPTIC FUNCTIONS, - - - 305 

APPENDIX, - - 340 
INDEX, 353 



INTRODUCTION. 



"L ETUDE approfondie de la nature est la source la plus 
feconde des decouvertes mathematiques. 

Non seulement cette etude, en offrant aux recherches un but 
determine , a Favantage d exclure les questions vagues et les 
calculs sans issue ; elle est encore un moyen assure de former 
1 Analyse elle-meme, et d en decouvrir les elements qu il nous 
importe le plus de connaitre et que cette science doit toujours 
conserves 

Ces ele ments fondamentaux sont ceux qui se reproduisent 
dans tous les effets naturels." (Fourier.) 

These words of Fourier are taken as the text of the present 
treatise, which is addressed principally to the student of 
ApjDlied Mathematics, who will in general acquire his mathe 
matical equipment as he wants it for the solution of some 
definite actual problem ; and it is in the interest of such 
students that the following Applications of Elliptic Functions 
have been brought together, to enable them to see how the 
purely analytical formulas may be considered to arise in the 
discussion of definite physical questions. 

The Theory of Elliptic Functions, as developed by Abel 

and Jacobi, beginning about 1826, although now nearly 
" *-~~^^*~ 

seventy years old, has scarcely yet made its way into the 



viii THE APPLICATIONS OF ELLIPTIC FUNCTIONS. 

ordinary curriculum of mathematical study in this country ; 
and is still considered too advanced to be introduced to the 
student in elementary text-books. 

In consequence of this omission, many of the most interest 
ing problems in Dynamics are left unfinished, because the 
complete solution requires the use of the Elliptic Functions ; 
these could not be introduced without a long digression, 
unless a considerable knowledge is presupposed of a course 
of Pure Mathematics in this subject. 

But by developing the Analysis as it is required for some 
particular problem in hand, the student of Applied Mathe 
matics will obtain a working knowledge of the subject of 
Elliptic Functions, such as he would probably never acquire 
from a study of a treatise like Jacobi s Fundamenta Nova, 
where the formulas are established and the subject is 
developed in strictly logical order as a branch of Pure 
Mathematical Analysis, without any digression on the 
application of the formulas, or on the manner in which 
they originate independently, as the expression of some 
physical law. 

In introducing these applications we are following, to some 
extent, the plan of Durege s excellent treatise on Elliptic 
Functions (Leipsic, Teubner); and also of Halphen s Traite 
des fonctions elliptiqucs et de leurs applications (Paris, 
1886-1891). 

But while volume I. of Halphen s treatise is devoted entirely 
to the establishment of the formulas and analytical properties 
of the functions, and the applications are not discussed till 
volume II. ; in the following pages it is proposed to develop 
the formulas immediately from some definite physical or 
geometrical problem ; and the reader who wishes to follow 
up the purely analytical development of the subject is referred 
to such treatises as Abel s (Euvres, Jacobi s Fundamenta Nova, 



INTRODUCTION. i x 

already mentioned, or the Treatises on Elliptic Functions of 
Cayley, Enneper, Kb nigsberger, H. Weber, etc. 

The following works also may be mentioned as having been 
consulted in the preparation of this work : 

Legendre: Theorie des f auctions elliptiques ; 1825. 
Thomas : Abriss einer Theorie der complexen Functionen 

und der Tketafunctionen einer Verdnderlichen ; 1873. 
Schwarz: Formeln und Lehrsdtze zum Gebrauche der 

elliptischen Functionen. 

Klein (Morrice) : Lectures on the Icosahedron ; 1888. 
Klein und Fricke; Vorlesungen uber die Theorie der ellip 

tischen Modalfunctionen ; 1890. 
Despeyrous et Darboux: Cours de niecanique ; 1886. 
R A. Roberts : Integral Calculus ; 1887. 
Bjerknes: Niels Hendrik Abel; tableau de sa vie et de son 

action scientifique ; 1885. 

We shall begin by the discussion of the Problem of the 

, as the problem best calculated to 



define the Elliptic Functions, and to give the student an idea 
of their nature and importance. 

Previously to the introduction of the Elliptic Functions, 
the Circular Pendulum could only be treated by means of the 
circular functions, by considering the oscillations as indefinitely 
small, and by assimilating its motion to that of Huygens 
Cycloidal Pendulum, of 1673. 

But now the employment of the Elliptic Functions renders 
the ordinary discussion of the Cycloidal Pendulum antiquated 
and of mere historical interest, and banishes from our treatises 
such expressions as " an integral which cannot be found," or 
"reducible to a matter of quadrature" in describing an elliptic 
integral, expressions which aroused the indignation of Clifford 
Mathematical Papers, p. 562). 



x THE APPLICATIONS OF ELLIPTIC FUNCTIONS. 

According to the new regulations for the Mathematical 
Tripos at Cambridge, to come into force in the examination 
in May 1893, the schedule II. of Part I. includes " Elementary 
Elliptic Functions, excluding the Theta Functions and the 
theory of Transformation " ; so it is to be hoped that this 
reintroduction of Elliptic Functions into the ordinary mathe 
matical curriculum will cause the subject to receive more 
general attention and study. These Applications have 
been put together with the idea of covering this ground by 
exhibiting their practical importance in Applied Mathematics, 
and of securing the interest of the student, so that he may if 
he wishes follow with interest the analytical treatises already 
mentioned. 

We begin with Abel s idea of the inversion of Legendre s 
elliptic integral of the first kind, and employ Jacobi s notation, 
with Gudermann s abbreviation, for a considerable extent at 
the outset. 

The more modern notation of Weierstrass is introduced 
subsequently, and used in conjunction with the preceding 
notation, and not to its exclusion ; as it will be found that 
sometimes one notation and sometimes the other is the more 
suitable for the problem in hand. 

At the same time explanation is given of the methods by 
which a change from the one to the other notation can be 
speedily carried out. 

It has been considered sufficient in many places, for instance 
in the reduction of the Integrals in Chapter II., to write 
down the results without introducing the intermediate analysis ; 
as the trained mathematical student to whom this book is 
addressed will have no difficulty in supplying the connecting 
steps, and this work will at the same time provide instructive 
exercises in the subject ; and further, in the interest of such 
students, many important problems have been introduced in 



INTRODUCTION. 



XI 



the text, forming immediate applications of theorems already 
developed previously. 

I have to thank Mr. A. G. Hadcock for his assistance in 
preparing the diagrams, and in drawing them carefully to 
scale. 



ERRATA. 

Page 6. Line 9 from bottom, read Huygeiis. 

42. Line 6, read sin" 1 */ -. 

V x-y 

48. Line 5 from bottom, read - 4tt-(9e- -r 4/r) J . 
64. Line 19, read Fonctions elliptiques. 

99. The diagram must be replaced by the one given below. 
The Xodoid in fig. 12, p. 99, was described by a point 
which was not a focus of the rolling hyperbola. 
107. Line 2 from bottom, delete minus sign before radical. 
138. Equation (7), read (r., 2 - ctf/D. 
158. Line 12, read 3QX(x, y). 
205. Line 6 from bottom, read $(u - v) - $>(u + v). 
213. Line 7 from bottom, read G + Lx - X(yz - y ~] - 
with the corresponding subsequent corrections. 
227. Line 7, read P s /- Y i + Q\ ; - Y 2 = - 
282. Line 5 from top, for rectangle read ribbon. 
328. Line 12 from bottom, read Pw. L. M. .?., IX. 




ABBREVIATIONS. 

Q. J. M., Quarterly Journal of Mathematics. 

Proc. L. M. S., Proceedings of the London Mathematical Society. 

Proc. G. P. $., Proceedings of the Cambridge Philosophical Society. 

Am. J. M., American Journal of Mathematics. 

F. E., Fonctions elliptiques (Legendre and Halphen). 

Math. Ann., Mathematische Annalen. 

Phil. Mag., Philosophical Magazine. 

Phil. Trail*. Philosophical Transactions of the Royal Society of London. 

Berlin Sitz., Sitzungsberichte der Berliner Akademie. 



CHAPTEK I. 

THE ELLIPTIC FUNCTIONS. 

1. The Pendulum; introducing Elliptic Functions into 
Dynamics. 

When a pendulum OP swings through a finite angle about 
a horizontal axis 0, the determination of the motion introduces 
the Elliptic Functions in such an elementary and straight 
forward manner, that we may take the elliptic functions as 
defined by pendulum motion, and begin the investigation of 
their use and theory by their application to this problem. 

Denote by W the weight in Ib. of the pendulum, and let 
OG = h (feet), where G is the centre of gravity ; let Wk 2 denote 
the moment of inertia of the pendulum about the horizontal 
axis through G, so that W(h* + k 2 ) is the moment of inertia 
about the parallel axis through (fig. 1). 

Then if OG makes with the vertical OA an angle 6 radians 
at the time t seconds, reckoned from an instant at which the 
pendulum was vertical ; and if we employ the absolute unit 
of force, the poundal, and denote by g (32 celoes, roughly) 
the acceleration of gravity, the equation of motion obtained 
by taking moments about is 



since the impressed force of gravity is Wg poundals, acting 
vertically through G ; so that 



or, on putting h + k*/h = I, 



G.E.F. 



THE ELLIPTIC FUNCTIONS. 




to 



THE ELLIPTIC FUNCTIONS. 3 

If the gravitation unit of force, the force of a pound, is 
employed, then the equation of motion is written 
w J lft 

fr/TO . 7 O\*-^ v/ TTTf " /\ 

(A. 2 +& 2 )-p = Wh sm 0, 
reducing to (1) as before. 



2. Producing OG to P, so that OP = l, GP = k 2 /h, the point 
P is called the centre of oscillation (or of percussion) ; and is 
called the length of the simple equivalent pendulum, because 
the point P oscillates on the circle AP in exactly the same 
manner as a small plummet suspended by a fine thread from 
(fig. 2); as is seen immediately by resolving tangentially 
along the arc AP = s = l9 ) when the equation of motion of 

the plummet is = g sin#= 



or I(d 2 0/dt 2 )= -g sin#; ........................ (1) 

and integrating, U(dO/dt) 2 = C-gversO ......................... (2) 

These theorems are explained in treatises on Analytical 
Mechanics, such as Kouth s Rigid Dynamics, or Bartholomew 
Price s Infinitesimal Calculus, vol. IV., and might have been 
assumed here ; but now we proceed further, to the complete 
integration of equation (2). 

3. First suppose the pendulum to oscillate, the angle of 
oscillation BO A +AOB being denoted by 2a (fig. 2); the angle 
of oscillation is purposely made large, as in early clocks, in the 
Navez Ballistic Pendulum, in a swing, or as in ringing a 
church bell, so as to emphasize the difference from small 
oscillations, the only case usually considered in the text 
books ; in fig. 2 the angle of oscillation is made 300. 
Then dO/dt = when 6 = a, so that in equation (2) 

(7=# versa ; 

and now denoting g/l by n 2 , so that n is what Sir W. Thomson 
calls the speed (angular) of the pendulum, 
^(dO/dt) 2 = n 2 ( vers a - vers 0) 

= 2?i 2 (sin 2 Ja-sin 2 ^), .............................. (3) 

since vers = 2 sin 2 J0 ; 

dO/dt = 27i x /(sin 2 ia -sin 2 i0), 

and nt-f ^ _ -.. . ..(4) 



4, THE ELLIPTIC FUNCTIONS. 

and (4) is called by Legendre an elliptic integral of the first 
kind ; it is not expressible by any of the algebraical, circular, 
or hyperbolic functions of elementary mathematics. 

4. To reduce this elliptic integral to the standard form con 
sidered by Legendre, we put 

sinA0 = sinJa sin 0, 

equivalent geometrically to denoting the angle ADQ by 
(fig. 2), where AQD is the circle on AD as diameter, touching 
BB f in D, and cutting the horizontal line PN in Q. 

For, in the circle AP, 



and, in the circle AQ, 

AN= \AD vers 2$ = AD sin 2 

= I vers a sin 2 = 21 sin 2 |a sin 2 0. 
Now sin 2 \ a sin 2 J$ = sin 2 |a cos 2 0, 

and J$ = sin ~ 1 (sin Ja sin 0), 

so that dle= .^i*.* *ft, v 

^/(l snrja sm 2 0) 

and therefore nt = fr- - r~r ^r-, > 

J ^/(1-siu 2 iasm 2 ^,) 



which is now an elliptic integral of the first kind, in the 
standard form employed by Legendre. 

(Fonctions Elli ptiques, t. I., chap VI.) 

5. In Legendre s notation, sin-Jet is replaced by ic; the quantity 
ic sin 2 <) is. denoted by A</> or A(0, K) ; and the integral 
ry(l-/c 2 sin 2 ^)-^ is denoted by F<f> or I\<J>,K), 



and called the elliptic integral of the first kind, being called 
the amplitude and K the modulus. 

Thus, in the pendulum motion, 

nt^Ffj), or F(</>, sinja). 

Legendre employs c instead of K, and puts K = sin 6 (a different 
^ to what we have just employed) and calls the modular 
angle ; and he has tabulated the numerical values of F((/>, K) for 
every degree of and 0. (Fonctions Elliptiques, t. II. Table IX.) 

Legendre spent a long life in investigating the properties of 
the function Fc/>, the elliptic integral of the first kind ; but the 
subject was revolutionised by the single remark of Abel (in 



THE ELLIPTIC FUNCTIONS. .5 

1823), that F(j> is of the nature of an inverse function ; and that 
if we put u F(f>, then we should study the properties of 0, 
the amplitude, as a function of u, and not of u as a function 
of <j>, as carried out by Legendre in his Fonctions Elliptiques. 

6. Jacobi proposed the notation = am u, or am(it, /c) when 
the modulus K is required to be put in evidence ; and now, 
considered as functions of u, we have Jacobi s notation 

cos < = cos am u, sin < = sin am u, A</> = A am u, 
the three elliptic functions of u\ and in Jacobi s Fundamenta 
Nova (1829) the properties of these functions, 
cos am u, sin am u, A am u f 

are developed, the elegance of Jacobi s notation tending greatly 
to the popularity of this treatise. 

7. Definition of the Elliptic Functions. 

Jacobi s notation is rather lengthy, so that nowadays, in 
accordance with Gudermann s suggestion (Tkeorie der Modular 
Functionen, Crelle, t. 18), cos am u is abbreviated to en it, 
sin am u to sn u, and A am u to dn u ; and 

en u, sn u, dn u 

are the three elliptic functions (pronounced, according to Hal- 
phen, with separate letters, as c, n, u ; s, n, u ; d, n, u) ; and they 
are defined by 

en u = cos 0, sn u = sin 0, dn u = A< = ^/(l /c 2 sin 2 <) ; 
where ^ is a function of u, denoted by am u, and defined by 
the relation 



so that 



/~amtt 

u = A/( 1 /c 2 sin 2 0) - $d<j> ; 





cZam u d(j> . -> 

and Y - = j r ^/ (I K" sm z (p) = dnit. 

rp , dcnu dcosd . dd> 

Thence , = , = sm0-^-= snudnu: 

du du * du 



and similarly 



d sn u d sin c?0 

7 - = 7 - = cos -f- = en u dn u ; 

du du ^ du 



n u <j> /rsn d> cos c< 

and , = ~r t -= ---- - ^ = /c 2 sn i^ en u 

du du A0 cZt6 




THE ELLIPTIC FUNCTIONS. 

8. Returning now with these definitions and this notation 
to the motion--o-~ilie^ pendulum, we have, on comparison, 
u = nt, while K = sin |<v\so that the modular angle is \a ; 



and K = AD/AB = AB/AE, K * = AD/AE (fig. 2) ; 

also ^ = am u, cos == en u, sin </> = sn u, d<f>/dt = n dn u ; 

d6/dt = 2nic cnu = 2 /i/c en nt, 

sin|$ = K sn u = K sn nt, 

cos J(9 = dn it = dn ?i ; 



AN = AD srfnt, ND = AD en 2 *, NE=AE dn*nt ; 



giving these quantities as elliptic functions of u or nt. 

9. We notice that ic = for infinitely small oscillations of 
the pendulum, the only case usually treated in the text- books ; 
and now <j> = u = nt, so that 

en u = cos u, sn u = sin u, while dn u = 1 ; 
and the elliptic functions have degenerated into the ordinary 
circular functions of Trigonometry. 

But in finite oscillations of the pendulum, where K is not 
zero, these new functions are required, which are called the 
elliptic functions; and their geometrical definition is exhibited 
in fig. 2, in a manner similar to that employed in Trigonometry 
for the circular functions. 

The name elliptic function is somewhat of a misnomer ; 
but arose from the functions having been first approached by 
mathematicians in their attempt at the rectification of the 
ellipse ( 77). 

For finite oscillations the circular functions are applicable 
only to cycloidal oscillations, as discovered by HuygKens, 1673, 
whence the motion on the arc of a cycloid is generally investi 
gated at length in elementary treatises ; but this discussion 
may be considered as of mere antiquarian interest, now that we 
are proceeding to discuss the finite oscillations of the pendulum 
by the aid of the elliptic functions. 

We may however make here a slight digression on cycloidal 
oscillations, treated in the manner we have employed for 
circular oscillations. 



THE ELLIPTIC FUNCTIONS. 



10. Gydoidal Oscillations. 

In the cycloid, fig. 4, the angle ADQ or <j> = nt (not &mnt, 
as in the circular pendulum) for all finite oscillations ; for 
as P oscillates on the arc BAB of the inverted cycloid 
described by the rolling of the circle AE, Q follows P at the 
same level on the circle AD with constant velocity. 




For if PQF meets the circle on AE as diameter in R, then, 
from a well-known property of the cycloid, the tangent TP is 
equal and parallel to A R, and half the arc AP ; and if n, p, q, r 
denote simultaneous consecutive positions of N, P, Q, R, 



the velocity of Q _ ^ Qq _ , Qq, Nn 
the velocity of P Pp ~~ Nn Pp 



= cosec qQP sin pPQ = cosec AFQ sin AER 

= _ IAN.AE 

N 



Now the velocity of P = ^(jlg . ND) 

and therefore the velocity of Q = ^AD Mti /(2g/AE) 
= AD^(g/l) = n . AD, a constant, 

if AE=\l; and therefore the angular velocity of Q about D 
is n, and the angle ADQ = <p = nt. 

Therefore the oscillations are isochronous, since the period 
is independent of the amplitude of oscillation. 



8 THE ELLIPTIC FUNCTIONS. 

But in the circular pendulum the period increases with the 
amplitude or angle of oscillation; because in the circle AP 
(fig. 2) the versed "sine AN varies as the square of the chord 
AP, while in the cycloid AP (fig. 4) the versed sine AN varies 
as the square of the arc AP. 

The time from P to A on the cycloid is equal to the c.m. 
(circular measure) of the angle ADQ divided by n or +J(gjl) ; 
and generally the time over any finite arc Pp of the cycloid 
will be equal to the c.m. of the corresponding angle QDq divided 
by n, supposing the body to start from the level of D. 

This will be true even when the point D is above E, as at 
D f , so that the body enters the cycloid with given velocity ; 
as for instance in the case of a railway train entering with 
given velocity V a cycloidal tunnel BAB under a river. 

Making DD = ^V 2 /g, the impetus of the velocity V, then 
the time occupied by the train in the tunnel from B to B is 
twice the c.m. of AD C divided by n. 

Also if the length of the tunnel is 2s, then s = ^/(2lh), if 
AD, the depth or versed sine of the tunnel, is h ; so that the 
time occupied is 

2, .DC II. , IAD 2s If h 






11. The Period of the Pendulum, and of the Elliptic 
Functions. 

The ^period of the pendulum is the name now given to 
the time of a double swing, according to the report of a Com 
mittee at the Conference of Electricians in Paris, 1889 : 
thus, if the swing is small, the period is 27r x /(/</) seconds. 

But if the angle, of vibration 2a is finite, the period is in 
creased ; denoting the period by T, and therefore the quarter- 
period, or time of motion of P from A to B (fig. 2) by %T, 
then as t increases from to \T, increases from to a, and 
from to J?r, so that nt or u increases from to K, where ( 4) 



and K (or F I K in Legendre s notation, and called by him the 
complete elliptic integral of the first kind) is now called the 
real quarter period of the elliptic functions, to the modulus K. 



THE ELLIPTIC FUNCTIONS. 
Now, expanding by the Binomial Theorem, 



z 

=1 

and, by Wallis s Theorem, 
/IT 
(sin 



Thus the period of a pendulum of length I, oscillating through 
. an angle 2a, is 



As a first approximation therefore in the correction for am 
plitude of swing, the period must be increased by the fraction 
J(sin |-) 2 of itself, or by 100(^ chord of a) 2 per cent. 

Thus a pendulum, which beats seconds when swinging 
through an angle of 6, will lose 11 to 12 seconds a day 
if made to swing through 8, and 26 seconds a day if made to 
swing through 10. (Simpson s Fluxions, 464.) 

The value of K or / V has been tabulated by Legendre 
for every degree and tenth of a degree in the modular angle 
(Fonctions Mliptiques, t. II., Table I.). 

We denote the modular angle by Ja, and put /c = sinja; 
while cosja is denoted by K arid called the complementary 
modulus, so that 

i 2 1 

K- + K ~ = 1 ; 

and then F I K is denoted by K , and called the complementary 
quarter period. 

The following table (from Bertrand s Calcul Integral, p. 714), 
gives the logarithms of the quarter periods /i" and -ZiTjCorrespond- 
ingtoeveryhalf degree in Ja, the quarter angle of swing; and then 

2/oc = sin a, /c = sinja, //^cosja, 
and ia is the modular angle. 

The modular angle in the Table is given from to 45 ; to 
determine K for a modular angle greater than 45, we look 
out the value of K corresponding to the complementary modu 
lar angle. 



10 



THE ELLIPTIC FUNCTIONS. 



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THE ELLIPTIC FUNCTIONS. 



11 



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(>l<^<^^^cc^rc?c^r^r^r^c^rc^^^^^^Tt < Tt T^ *^i- ri ; -:L :o 

b 

Lt O u-5 O & O LO O ^ O U2 O O O O O O O O Ut O >--: O tr: O O O O 

*.l* 

re u-t oo ^ 

l5c?|8Sc|c?|fS?222?S2?:2?85lSSS|l8|, 



12 THE ELLIPTIC FUNCTIONS. 

12. We notice that when the modular angle is 15, then 
log K lK =-2385606=1 log 3, so that K /K=^3: 

this will be proved subsequently ; but it shows here that the 
period of a pendulum oscillating through 300 is ^/3 times the 
period when the pendulum oscillates through 60. 

Again we shall prove subsequently that, 
if K /K=J t 7,iheiL2icK = l , 

so that equal parallel horizontal chords, BE the higher, and 
W the lower, each of length one-eighth the diameter, cut off 
arcs of the circle below them, which would be swung through 
by the pendulum in times which are in the ratio of ^/l to 1. 

Many other similar numerical examples can be constructed 
when the Theory of the Complex Multiplication of Elliptic 
Functions is studied. 

13. When a=j7r, the pendulum drops from a horizontal 
position and swings through two right angles, as in the Navez 
Electro-Ballistic Pendulum ; and now K=K , and the modular 
angle is JTT. 

Table II. from Legendre s Fonctions Elliptiques, t. II., gives 
to five decimals the value of u = F<f> for every half degree in 
the value of <, when the modular angle is 45 ; and thence by 
means of the preceding formulas which determine the motion 
of the pendulum by elliptic functions, the pendulum can be 
graduated so as to measure small intervals of time A = Au/%, 
as required for electro-ballistic experiments. 

Then from Table II., when K = K , and K = K=$J2, 
en u = cos <, sn u = sin <, dn u = ^/(l J win 2 ^). 

14. Generally in the pendulum, K=^nT, so that the period 



When /c = 0, K=\ir, and the period is ^^(l/g), as proved 
otherwise in the ordinary elementary treatises, for small 
oscillations of the pendulum^ 

But in the finite oscillations of the pendulum, with 



then ( 8) dO/dt = ln K en 4-Kt/T, 

,sin-0= K 



Putting = 0, 16 = 0, we find 

cnO = l, snO = 0, dnO=l ; 



THE ELLIPTIC FUNCTIONS. 13 

and putting t = J7 7 , u = K, = JTT, 

when the pendulum has swung to OB, 

en A" = cos JTT = O, sn K=l, dn K = K ; 
while putting t = \T, u = 2 A", 

when the pendulum is swinging backwards through the verti 
cal OA, cu2K=-l,sn2K=0, dn2A =l; 

analogous to the values of cos and sin 0, for $ = 0, ^TT, TT ; 
so that 2K is the Aa/ period of the elliptic functions, corre 
sponding to the half period TT of the circular functions. 

rir ( p rir /-$ 

Since /c?0. A< = Yd0/A0 /d<f>/&<f> = 2K u, if = am u, 

000 

therefore am(21Ttt)= -jr<p= 7ramt; 

and generally am(2 m A" u) = m?r = m?r am ^6 ; 
so that cn(2??iJ*ri6) = COS(??ITT am IL) = ( 1 ) y cn u, 

sn(2mKu) = sin(m7rain u)=( l) m sn u, 
while dn(2m-fiT it) = dn u : 

analogous to cos(m?r 6)= ( l) m cos 0, 
sin(m7r#) = ( - l) ?n sin ; 
and representing the motion, ??i half periods, past or future. 

15. The degenerate Circular and Hyperbolic Functions. 

As a increases from to TT, K increases from to 1, and K 
from |-TT to infinity; the pendulum has now, with /c = l, just 
sufficient velocity to carry it to the highest position, and this 
will take an infinite time. 

For with a = ?r, equation (3), page 3, becomes 
I (dO/dt) 2 = rr(l + cos 0) = 2n 2 cos 2 J0 ; 



= log tan JO + 6) = log(sec J0 + tan J0), 
which is infinite when O = TT. 

In Small oscillations the period is 27r/n, and the motion of 
M, the projection of P on the horizontal axis Ax, is then a 
Simple Harmonic Motion (s.H.M.) given by the differential 

d 2 x 
equation _ + n -x = 0, 

the solution of which is 

o; = .4cos^, or Bsmnt, or A cosnt+Bsmnt, or acos(7i^-f e) ; 
so that n is the constant angular velocity round D of the point 
Q on the infinitesimal circle AQD, as in the cycloid. 



14 THE ELLIPTIC FUNCTIONS. 

In Kepler s Problem in Astronomy, n represents what is 
called the mean motion of a planet or satellite, and nt ornt+e 
the mean anomaly ; a satellite of Jupiter, when observed in 
the plane of its orbit, supposed circular, will appear to move 
with a s. H. M. 

But with *==!, putting J0 = = angle AEP (fig. 3) 
nt =ysec 0^0 = log(sec + tan 0), 



so that sec + tan <f> = e nt , 
sec tan = e~ w< , 
sec = \(e ni + e~ nt ) = cosh nt, 
tan = (e nt - e ~ nt ) = sinh nt, 
sin (f> = tanh nt, cos = sech nt, 
tanJ0 = tanhJ?i, and so on. 
Also dO/dt = 2n cos J0 = 2n sech nt ; 

so that if the angular velocity of the pendulum in the lowest 
position OA is 2n, the pendulum will just reach the highest 
position OE ; but the time occupied in reaching it will be in 
finite, since 6 = 7r, 0=.j7r makes nt and therefore t infinite. 
The velocity of P in any position is 

l(dO/dt) = 2nl cosW = n . EP, 
and therefore varies as EP. 

If EP in fig. 3 is produced to meet Ax in M t then 
AM = AE tan0 = 21 sinh nt, EM = EA secJ0 = 21 cosh nt ; 
so that, if AM or EM is denoted by x, 

d2x -n*x-0 
~dt* 

the general solution of which differential equation is 

x A cosh nt-\-B sinh nt. 

16. When the pendulum just reaches the highest position 
OE, ic = l ; and u (or nt) and 0, the c.m. of the angle AEP, 
are connected by the relations 

u =y*sec d<j> = log (sec <f> + tan 0) 

= cosh ~ ^ec $ = sinh ~ Han <p = tanh - 1 sin = 2 tanh ~ : tan J0. 

Conversely 

= cos ~ 1 sech u = sin ~ a tanh u = tan ~ J sinh u = 2 tan ~ Hanh Ju ; 
and then is called by Professor Cayley the Gudermannian 
of u, and denoted by gdu; so that if = gdu, then 

u = gd " J = log (sec + tan 0) = cosh - ^ec 0, etc. 



THE ELLIPTIC FUNCTIONS. 15 

Holiel proposes for <p the name of hyperbolic amplitude of 
16, with the notation = amh u, instead of gd 16 ; so that 



amh u 

sec <>d 



16=/S< 

o 

or ^ = arnh u = /sech udu = cos ~ 1 sech 16 = sin Hanh u, etc ; 



analogous in the general case of the elliptic functions, for any 

modulus /c, to ( 7) 



= / 



u, etc. 



As degenerate forms, when K = 1 , 

en it = sech u, sn 16 = tanh u, dn u = sech u ; 
while, with /c = 0, 

en u = cos 16, sn u = sin it, dn u = 1 . 

Thus, when /c = l, the elliptic functions degenerate into the 
hyperbolic functions ; and, when K = 0, into the circular func 
tions ; but with any other value of the modulus K, the elliptic 
functions must be considered as new functions, of a higher 
order of complexity than the circular or hyperbolic functions. 
The following Table, from Legendre, F. E., t. II., Table IV., 
gives the values of 

16 = log (sec 4- tan </>) = log tan( JTT + J0) 

for every degree of radians ; whence the numerical values of 
the hyperbolic functions of u can be determined, by aid of a 
table of circular functions, and by the relations 

cosh u = sec 0, sinh u = tan <p, tanh u = sin 0, .... 
For values of u greater than about 4 the Table fails ; but 
then it is sufficient, to two decimals, to take 

cosh u = sinh u = Je M ; 
Iog 10 cosh 16 = Iog 10 sinh u = Mu log 2 ; 
or, to a closer approximation, 

Iog 10 cosh u = MIL log 2 -f Me ~ 2u , . . . , 
Iog 10 sinh u Mu log 2 Me ~ 2 ", . . . , 
Iog 10 tanhu= -2 Me~ 2u ..., 

M denoting the modulus Iog 10 e. 

(Proposed Tables of Hyperbolic Functions, Report to the 
British Association, 1888, by Prof. Alfred Lodge.) 



16 



THE ELLIPTIC FUNCTIONS. 



TABLE TIL 








u 







u 







M 





o-ooooo 


o-ooooo 


30 


0-52360 


0-54931 


60 


1-04720 


1-31696 


1 

2 

3 


0-01745 
0-03491 
0-05236 


0-01745 
0-03491 
0-05238 


31 
32 

33 


0-54105 
0-55851 
0-57596 


0-56956 
0-59003 
0-61073 


61 
62 
63 


1-06465 
1-08210 
1 -09956 


1 -35240 
1-38899 
1-42679 


4 
5 
6 


0-06981 
0-08727 
0-10472 


0-06987 
0-08738 
0-10491 


34 
35 
36 


0-59341 
0-61087 
0-62832 


0-63166 
0-65284 
0-67428 


64 
65 
66 


1-11701 
1-13446 
1-15192 


1-46591 
1-50645 
1-54S55 


7 
8 
9 


0-12217 
0-13963 
0-15708 


0-12248 
0-14008 
0-15773 


37 

38 
39 


0-64577 
0-66323 

0-68068 


0-69599 
0-71799 
0-74029 


67 
68 
69 


1-16937 
1-18682 
1 -20428 


1-59232 
1 -63794 
1-68557 


10 


0-17453 


0-17543 


40 


0-69813 


0-76291 


70 


1-22173 


1-73542 


11 
12 
13 


0-19199 
0-20944 
0-22689 


0-19318 
0-21099 
0-22886 


41 
42 
43 


0-71558 
0-73304 
0-75049 


0-78586 
0-80917 
0-83284 


71 
72 
73 


1-23918 
25664 
27409 


1-78771 
1-84273 
1-90079 


14 
15 
16 


0-24435 
0-26180 
0-27925 


0-24681 
0-26484 
0-28295 


44 
45 
46 


0-76794 
0-78540 
0-80285 


0-85690 
0-88137 
0-90628 


74 

75 
76 


29154 
30900 
32645 


1-96226 
2-02759 
2-09732 


17 
18 
19 


0-29671 
0-31416 
0-33161 


0-30116 
0-31946 
0-33786 


47 
48 
49 


0-82030 
0-83776 
0-85521 


0-93163 
0-95747 
0-98381 


77 
78 
79 


34390 
36136 

3788 1 


2-17212 
2-25280 
2-34040 


20 


0-34907 


0-35638 


50 


0-87266 


1-01068 


80 


1-39626 


2-43625 


21 
22 
23 


0-36652 
0-38397 
0-40143 


0-37501 
0-39377 
0-41266 


51 
52 
53 


0-89012 
0-90757 
0-92502 


1-03812 
1-06616 
1-09483 


81 
82 
83 


41372 
43117 
44862 


2-54209 
2-66031 
2-79422 


24 
25 
26 


0-41888 
0-43633 
0-45379 


0-43169 
0-45088 
0-47021 


54 
55 
56 


0-94248 
0-95093 
0-97738 


1-12418 
1-15423 
1-18505 


84 

85 
86 


46608 
48353 
50098 


2-94870 
3-13130 
3-35467 


27 

28 
29 


0-47124 

0-48869 
0-50615 


0-48972 
0-50939 
0-52925 


57 
58 
59 


0-99484 
1-01229 
1-02974 


1-21667 
1-24916 
1 -28257 


87 
88 
89 


51844 
53589 
1-55334 


3-64253 
4-04813 
4-74135 


30 


0-52360 


0-54931 


60 


1-74720 


1-31696 


90 


1-57080 


infinite. 



THE ELLIPTIC FUNCTIONS. 



17 



Considered. as a function of the latitude <, u was called the 
meridional part by Edward Wright, 1599, who first employed 
it for the accurate construction of the parallels of latitude on 
the Mercator Chart, by making the ratio of the distance from 
the equator of the parallel of latitude < to the distance between 
the meridians whose difference of longitude is equal to the 
ratio of u/<j> ( 98). 

17. Returning to the general elliptic functions, we notice 
that en 2 u + sn 2 u = l, 

dn 2 u -f- /c 2 sn 2 w, = 1 , 



or, in a tabular form, 



en 



sn 



dn 



en u 



7(1-811%) 

snu 



dnu 



whence any one of the three elliptic functions en, sn, dn, can 
be expressed in terms of any other ; the three functions are 
thus not absolutely necessary, but all three are retained and 
utilized for simplicity of expression, as sometimes one and 
sometimes another is most appropriate for the particular pro 
blem in hand ; in the same way, of the circular functions 

cos 9, sin 9, tan 0, cot 9, sec 9, cec 0, vers 9, 
one would be sufficient, but all are useful ; and so also with 
the hyperbolic functions cosh u, sinh u, tanh u, 

For the reciprocals and quotients of the elliptic functions 
en, sn, dn, a convenient notation has been invented by Dr. 
Glaisher, according to which 1/cn u is represented by nc u, 
1/sn u by ns u, 1/dn u by nd u, en u/dn u by cd u, and so on. 

In this manner sn u/cn u would be denoted by sc u ; but it 
is more commonly denoted by tanam u, abbreviated to tn u ; 
while en u/sn u or cs u would be denoted by cotam u, or ctn u. 
, According to Clifford (Dynamic, p. 89) we might abbreviate 
the designation of the hyperbolic cosine, sine, and tangent to 
he, hs, and ht ; or we may write them ch, sh, th ; with en, sn, 
tn for the elliptic functions ; and merely c, s, t for the circular 
functions. 



G.E.F. 



18 THE ELLIPTIC FUNCTIONS. 

18. Pendulum performing complete revolutions. 

Secondly, suppose the pendulum performs complete revolu 
tions (fig. 3). 

We have seen previously ( 15) that if the pendulum has 
an angular velocity 2n = 2^/(g/l) in the lowest position, it 
will just reach the highest position; and therefore if this 
angular velocity is increased, the pendulum will perform com 
plete revolutions. 

The integration of equation (1) in the form 



or ^v 2 /g + ANAD, a constant, denoted by 2R, 

shows that the velocity of P is that which would be acquired 
in falling freely from the level of a certain horizontal line 
BDB , which now does not cut the circle, as in fig. 2 when the 
pendulum oscillated, but lies entirely above the circle, as in 
fig. 3, at a height 2R above the lowest point A ; and the im 
petus of the velocity of P is the depth of P below BB . 
Denoting the angle AEP by 0, so that = |$, then 
2l 2 (d<p/dt) 2 = g(2R I vers 20) == 2g(R . 

/cZ0\ 2 g 
or l-^-J = - 

on putting /c 2 = l/R = AE/AD ; and n 2 =g/l, as before ; 

so that nt/ K =f(l - K 2 sin 2 0) - cZ0 = P(0, K\ 



in Legendre s notation ; and inverting the function according 
to Abel s suggestion, with Jacobi s notation, 

and now, with Gudermann s abbreviated notation, 
cos ^0 = en. nt/K, 

d0 ^n , ,, 

di = K /* 

AN= I vers = 2Z sin 2 = AEsn 2 nt/ K , 
NE AE cri*nt/K, ND = AD dn 2 nt/K, 
AP = AE sn nt/K, PE=AE en nt/ K , 
NP = 21 sin ^0 cos J$ = AE sn nt/K en nt/K. 



THE ELLIPTIC FUNCTIONS. 19 

19. The time of moving from A to E is obtained by putting 
= IK, and is therefore Kic/n ; and therefore the period, or 
time of a complete revolution, is 2KK/n (not 4jfic/n). 

With the series for K as given in 11, and with K 2 
the period of the pendulum for a complete revolution is 



The analogous expression for the period when the pendulum 
oscillates, rising on each side to a height 2R, less than 21, is, 
as in 11, 

8 



Putting /c = 1, and R = I, makes K infinite, and brings us back 
again to the separating case between oscillations and complete 
revolutions of the pendulum ; and we thus regain for this 
case the original expressions involving hyperbolic functions, 
previously investigated in 15. 

But as K now diminishes again from 1 to 0, the pendulum 
revolves faster and faster, until finally, when Ac = 0, we must 
suppose the pendulum to revolve with infinite angular velocity, 
the fluctuations of which for different positions of P are in 
sensible ; and the period is now zero. 

20. We notice that, in the circle AQ (fig. 2) the point Q 
moves according to the law 

^ = am nt, 

so that Q moves round in a circle, centre 0, in fig. 2 like the 
point P making complete revolutions in fig. 3. 

But now, in the motion of Q, gravity must be supposed 
diluted from g to K 4 g ; for if R or K-l denotes the radius of the 
circle A Q, g the diluted value of gravity, and n = *J(g /R) the 
speed of the pendulum CQ, then we must have 

$ = am nt = am rit/K, 
so that n = Kn 



We may dilute gravity in the circle A Q by inclining the 
plane of the circle to the vertical at an appropriate angle. 



20 THE ELLIPTIC FUNCTIONS. 

21. Another way of diluting gravity would be to replace the 
circle AQ by a fine tube in the form of a uniform helix with 
horizontal axis through its centre G perpendicular to the plane 
of the circle AQ, and to suppose the particle Q to move in this 
helix under gravity. 

Then we shall find that if the length of one complete turn 
of this helical tube is equal to the circumference of the circle 
AP, the particle Q moving with velocity due to the level of E 
will follow the motion of the particle P moving on the circle 
AP with velocity due to the level of D, so that PQ will always 
be horizontal, if once it is horizontal, and P, Q will always be 
at the same level during the motion. 

For in this case the mechanical similitude is secured by in 
creasing the square of the velocity of Q in the ratio of 1 to 
l//c 4 , instead of diluting gravity to /c 4 </. 

We may secure the same effect by supposing Q to be a point 
on a pendulum CQ , of length greater than GQ ; or else of length 
GQ, but of which the axis (7 is cut into a smooth screw of 
appropriate pitch ; or else engaging with teethed wheels, so as 
to increase the angular inertia about G. 

22. If we produce GQ to any fixed distance CQ =l , then Q 
will also perform complete revolutions like a pendulum of 
length , with gravity changed in a certain fixed ratio depend 
ing on I ; and we can keep gravity unchanged by choosing I 
so that n 2 = g/l = K V = K 2 g/l, 

or I = l/ K * = lcosec 2 a; 

and now Q revolves with velocity due to a level at a height 
2//c 4 = 2cosec 4 Ja above its lowest position; so that the period of 
revolution of a simple pendulum of length I cosecHa, when the 
velocity is due to the level of a line at a height 2cosec 4 Ja above 
its lowest point is equal to the time of oscillation of a simple 
pendulum of length I through an angle 2 from rest to rest. 

These problems on the pendulum have been developed here 
at some length, in accordance with the idea of this Treatise, 
that it is simple pendulum motion which affords the best 
concrete illustration of the Elliptic Functions. 

Similar principles are involved in the following three 
theorems, which the student can prove as an exercise in the 
manner employed for the cycloid in 10. 



THE ELLIPTIC FUNCTIONS. 21 

1. If two vertical circles, of diameters AD and AE, touch at 
their lowest points A, the time of oscillation from rest to rest 
of a particle in the circle AE with velocity due to the level 
of D will be to the time of revolution of a particle in the 
circle AD with velocity due to the level of E in the ratio of 
AEio AD (fig. 2). 

2. Two particles move, under gravity, in vertical circles. 
The one oscillates ; the other performs complete revolutions. 
Prove that if the height to which the velocity of the first is due 
bears to the diameter of the first circle the same ratio as the 
diameter of the second circle bears to the height to which the 
velocity in it is due (the heights being measured from the low 
est points of the circles) the ratio of the squares of the times 
in corresponding small arcs and therefore the squares of the 
whole times of oscillation and revolution will be that com 
pounded of either of the before-mentioned equal ratios and 
the ratio of the diameters of the circles. 

3. Two equal smooth circles are fixed so as to touch the same 
horizontal plane, their planes being at different inclinations ; 
two small heavy beads are projected at the same instant along 
these circles from their lowest points, the velocity of each bead 
being that due to the height of the highest point of the other 
circle above the horizontal plane, show that during the motion 
the two beads will always be at equal heights above the hori 
zontal plane. 

23. We have compared the motion of the pendulum in fig. 1 
with that of the simple equivalent pendulum composed of 
the particle P moving on a smooth circle, or at the end of a 
fine thread or wire OP ; oscillating from B to E in fig. 2, and 
performing complete revolutions in fig. 3, the velocity of P at 
any point being that acquired in falling from the level of D. 

Taking as coordinate axes the horizontal and vertical axes 
Ax and Ay through A, and referring the motion of P to the 
coordinates x and y, then since P describes the circle AP of 
radius I, x 2 = 2ly y 2 . 

Denoting by v = ds/dt the velocity of P, then by the principle 
of energy i^ 2 /# = 2P y, 

2R denoting the height of D above A. 



22 THE ELLIPTIC FUNCTIONS. 

dx l 

But since -- 



_ dx 2 _ V 
*~ +2 ~- 



while (ds/dt) 2 = g(2R - y) ; 

so that l 2 (dy/dt) 2 =g(2R- y)(2ly - 

dt I 




called an elliptic integral in /, and of the ^rs^ kind. 

24. Firstly, if the pendulum oscillates, R is less than I, and 
2/ oscillates between and 2R ; and the integral is reduced to 
Legendre s canonical form by putting y = 2R sin 2 ; when 

nt =(! - K 2 sin 2 ) - 4d = ^(0, /c), 



where K 2 = R/l, n 2 =g/l , 

and therefore with Jacobi s and Gudermann s notation, 



and y = 2J? sn% = 2^/c 2 sn 2 ?i^ a; = 2fo sn ^ dn nt ; 

or ^=^Dsn 2 ^, ND = ADcn 2 nt, NE=AjEdu 2 nt, 
as before, in 8. 

25. When /c = 0, the oscillations are indefinitely small ; 
and now y = 2^ sin 2 ?i, 
where R is a very small quantity ; 

and nt = I - - vr = sni ~ 1 A/ Jt 

- ^2R 



an ordinary circular integral. 

It was Abel who pointed out (about 1823) that in looking 
only at the Elliptic Integrals, mathematicians had been taking 
the same difficult point of view as if they had begun to deduce 
the theorems of elementary Trigonometry from an examination 
of the properties of the inverse circular functions, as deduced 
from the circular integrals. 

(Niels-Henrik Abel. Tableau de sa vie et de son action 
scientifique. Par C. A. Bjerknes. 1885.) 



THE ELLIPTIC FUNCTIONS. 23 

26. Secondly, if the pendulum performs complete revolu 
tions, as in fig. 3, R is greater than I, and y oscillates in value 
between and 21 ; we now reduce the elliptic integral in 23 
to Legendre s standard form by putting y = 2l sin 2 0, 

when nt/K=f(lK 2 sin 2 0)"W^=^[^, K) 



where K 2 = l/R } 

the reciprocal of its former expression ; and now 

= &m(nt/K, K), y = 2l sn?nt/K, x = 2lsii nt/K en nt/K ; 
or AN=AEsu 2 nt/ K , NE=AEcu 2 nt/ K , ND = AD dn*nt/ K , 
as proved before, in 18. 

27. In the separating case between oscillations and complete 
revolutions, R = l, and now K = 1 ; 

and y = 21 sm 2 = I vers2< = I vers ; 

also ( 23) nt =ysec Qd^ = log(sec <p + tan 0) 

= cosh " 1 sec = sinh ~ Han = tanh ~ x sin = 2 tanh ~ Han J< ; 
so that < = gd nt, or arnh nt, 

and sec = cosh nt, tan = sinh nt, sin = tanh nt, 

y = 21 tanh 2 ?i, x = 21 sech nt tanh nt, 
as before, in 15. 

28. Landen s Point. 

With centre E in fig. 2 and radius j5 describe a circle 
cutting the vertical AE in L ; then Z is an important point in 
the theory of pendulum motion and elliptic functions, called 
Landen s point. 

Since EB* = ED.EA= EC 2 - CA\ 

therefore the circle, centre E and radius EB, will cut the circle 
AQD, centre C, at right angles ; and 



since LC 2 +CQ 2 = LC- 2 +EC 2 -EL*=,2LC . EC, 

and EL = EB = 21 K , EC=l(l+ K ^, LC=l(l- K J. 
Now, by 20, the velocity of Q 

= J(2g . EN) = J(2g K * . ^Y) = n^(2l . EN) 
=n.LQ(I+ K ). 

Similarly in fig. 3, where P makes complete revolutions, the 
velocity of P = n.LP(I+K)/K, where the Landen point L is 
obtained by drawing a circle with centre D, cutting the circle 
^.^orthogonally, and the vertical AD in L. 



24 THE ELLIPTIC FUNCTIONS. 

We shall prove subsequently that any straight line through 
L divides the circle APE in fig. 3 (or the circle AQD in fig. 2) 
into two parts, each described in half the period. 

29. Change from one modulus to its reciprocal. 

It is important for the simplicity and for convenience of 
tabulation of the elliptic functions that the modulus K should 
not exceed unity ; but the preceding reductions of the motion 
of the pendulum to elliptic functions, in the two cases in which 
the pendulum oscillates and performs complete revolutions, 
show us how to make the elliptic functions to a modulus /c, 
which is greater than unity, depend on the elliptic functions 
to the reciprocal modulus l//c, which is less than unity. 

For, on comparing the two expressions for y, according as 
the pendulum oscillates or performs complete revolutions, 

y = 2Rsn\nt, K ), or 2lstf( K nt, l//c), 
where /c 2 = Rjl ; 

so that K 2 sri z (nt, K) = sn 2 (V?i, I/K) ; 

or, putting nt = u, 

K sn(u, K) = sn (KU, I/K), 

so that dn(u, /c) = en (KU, l//c), 

cn(^, /c) = dn (KU, I/K). 

Independently, if we suppose < = am(^, /c), and if we put 

K sin (j> sin i^, 

then K cos <pd<j> = cos \/r d\js, 

and cos 



so that u = /(I /c 2 sin 2 </>)~*<i0 /s 

u 

/s 



KU = sec 



or 



and since /c sin = sin i/r, etc., 

therefore K sn(w., /c) = SV.(KU } l//c), etc. 

When u = K, = j7r, and i/r^sin" 1 ^; so that, if K is less 

/sin" 1 * 
( 1 /c ~ 2 sin 2 \/r 



THE ELLIPTIC FUNCTIONS. 25 

30. Rectilinear Oscillations expi^essed by Elliptic Functions. 
In simple pendulum motion, referred to horizontal and ver 
tical axes Ax, Ay, drawn through the lowest point A, we have 
shown in 24, 26, that 

y = 2lK 2 su 2 nt, x = 2//c sn nt dn nt ; 
or y = 2lsn?nt/K f x = 2lsn nt/K en ntJK ; 

according as the pendulum oscillates or performs complete 
revolutions. 

Treating the vertical motions separately, and differentiating 
according to the rules established in 7, we find, on taking 

y = 2/c 2 sn 2 ?i, 

dy/dt = 4ilnK 2 sii nt en nt dn nt 
d 2 y/dt 2 = 4<ln 2 K\cn?nt dn-nt - sn 2 nt dn 2 ?? t - K 2 su 2 nt ctfnt) 



--+ , by 17. 
I IK." 46-/C-/ 

Taking y = 2lsn 2 nt/K, we find in a similar manner 



both immediately obtainable from the equation of 23, 



whence I 2 (d 2 y/dt 2 ) = g(Rl -Ry~ly+ f^ 

We shall find similar expressions for d-y/dt 2 when y varies 
as cn 2 ?i or dn. 2 nt, all of the form 



Let us determine then, as exercises in the differentiation of 
the elliptic functions, the acceleration d 2 x/dt 2 , and thence the 
force at a distance x t which will make a body oscillate in a 
straight line according to one of the laws 

x = a en nt, sn nt, dn nt, tn nt, nc nt, ns nt, .... 
Taking x = a en nt, 

dx/dt na sn nt dn 7i 
d 2 x/dt 2 ri 2 a(cn nt dn 2 nt K 2 su 2 nt en nt) 

Ml 



26 THE ELLIPTIC FUNCTIONS. 

so that jp 

reducing to zero when /c = 0. 

It is often simpler to find dx/dt, and then to express 
as a function of x ; and then a differentiation with respect to t 
will give d 2 x/dt 2 immediately as a function of x. 

Thus, if x = a sn nt, 

dx/dt = na en ?i dn nt 



80 that 



reducing to zero, when /c = 0. 
Similarly, if # = a dn 71*, 



Generally, when cc varies also as tn ?i, nc TI, . . . , we shall 
find a relation of the form 



which, when multiplied by dx/dt and integrated, gives 

KdxJMf = 

or dx/dt = 



an elliptic integral, of which the different expressions are given 
in Chapter II. 

31. A Special Minimum Surface. 

Another interesting exercise in the differentiation of elliptic 
functions is to verify that the surface discovered by Schwarz 
(Gesammelte Mathematische Abhandlungen, vol. I., p. 77), 

cnx+cny + cnz + cnx cuy en = 0, 

with the modulus /c = J, is a minimum surface, having zero 
curvature at every point, and therefore satisfying the condition 



p, q, r, s, t having their usual meaning as partial differential 
coefficients of s with respect to x and y. 



THE ELLIPTIC FUNCTIONS. 27 

Schwarz shows that this condition is equivalent to 



p v p. 2 denoting the principal radii of curvature of the surface 
(C. Smith, Solid Geometry, 255), where 

Y- P v = q 

J(p 2 + q 2 + lY J(f+f+l) 

Let us write c v s v d v for en x, sn x, dux-, and c 2 , s. 2 , d 2 , C 3 , S 3 , 
d 3 for the same functions of y and z. 

Then c x + c. 2 + c 3 + c^Cg = ; 

and differentiating with respect to x, 



But 



(l+c lC2 ) 2 
so that s s (l + CiC 2 ) = s^, etc. ; 

_vA = __?A = r 



-,-. 

By symmetry, g= ___ 

so that we may write 



where D = (c^ 8l ) 

S 

By symmetry 



o 
2 3 



p, y p, T r- 

so that -t-- =0, provided that 



28 THE ELLIPTIC FUNCTIONS. 




or 

or, since s x 2 = 1 - c^, d* = i(3 + q 2 ), 



or (<?! + c 2 + c s + c^) (3 - c 2 c 3 - c 3 c x - c^) = 0, 

and this is true, in consequence of the original relation 
Ci-f-Cg+Ca+CiCjCjrsO. 

The other relation 3 c 2 c 3 c^ c x c 2 = 
represents isolated conjugate points, where 

C 1 = C 2 = C 3 = 1 - 

Another minimum surface is 

tn y tn 2 + tn s tn 03 + tn x tn ^/ + 3 = 0, 

With K 



32. Elliptic Function Solution of Euler s Equations of 
Motion. 

Before leaving the mechanical interpretation of elliptic 
functions, we may just mention here an important application, 
the application to the solution of Euler s equations of motion, 
for a body under no forces, moving about its centre of gravity, 
or about any fixed point. 

Euler s equations for p, q y r, the component angular velocities 
about the principal axes, are (Routh, Rigid Dynamics) 



where A, B, C denote the moments of inertia about the princi 
pal axes ; and two first integrals of these equations are 

= T, a constant ; 

=G 2 , a constant, 
obtained by multiplying Euler s equations respectively by (i.) 
p, q, r, and adding, (ii.) by Ap, Bq, Or, and adding ; and then 
integrating. 

Comparing these equations with the equations of 7, 

cn w = sn- u dn u, sn u = en u dn u, dn u = /c 2 sn u en u, 
where accents denote differentiation with respect to u, we 
notice that if A > B > C, and the polhode includes the axis (7, 
so that AT>BT>G 2 >CT, we may put u = nt, and 



THE ELLIPTIC FUNCTIONS. 29 

p = P cn u, q = Q sn u, r = RdnU , 

and then, on substituting in Euler s equations of motion, 
B-G_nP A-C_nQ A-B_ K 2 nR 
A ~QR B ~RP C = PQ 
Putting t = 0, and therefore p = P, q = 0, r = R; then 
AP* + CR 2 = T, A 2 P 2 + C 2 R 2 = G\ 

G*-CT AT-G 2 

so that P ~ 

,., n , 

and then 



L B B-C~B(B-CY 
while n 2 = R 2 



and 



AB ABC 

P 2 A A-B_G*-CTA-B 
& c B-C~AT-G 2 B-C 



If the polhode encloses the axis of greatest moment A, so 
-that AT > G 2 > BT > CT, we must put . 

2) = P dn u, q = Q sn u, r = Rci].u; 
and then determine P, Q, R, n, K as before ; when 

AT-G 2 B-C 



2 _- 9 _ 

ABC ~G 2 -CTA-B 

In the separating case, when G 2 = BT, then /c = l, and 

p = P sech nt, q = Q tanh nt , r = R sech nt ; 
so that, when t = 0, 

2 _ G 2 B-C _ . J _G 2 A-B 
P ~ABA-C q ~ ~BCA-C 
and initially or finally, when t = + yz , 

2 ) = 0, q= G ! B, r = 0; 
and the body is spinning about its mean axis B. 

But when the body is spinning about the axis of greatest or 
least moment, G 2 = A T= A 2 p 2 , or G 2 = CT= C 2 r 2 , and K = ; and 
the period of a small oscillation is 2-Tr/n, where 

. (A-B)(A-O (A-B)(A-0) 

ABC. BC P 



_ 

We shall return subsequently to these equations in Chap. III. 



CHAPTER II. 

THE ELLIPTIC INTEGRALS (OF THE FIRST KIND). 

33. In Chapter I. we have immediately made use of Abel s 
valuable idea of the Inversion of the Elliptic Integral, which 
is the foundation of the modern theory of the Elliptic Func 
tions ; and we have considered the functions which are inverse 
to the elliptic integral, and treated them as the direct funda 
mental functions of our Theory. 

Previously to Abel s discovery (1823) it was the elliptic 
integral which was studied, as in the writings of Euler and 
Legendre ; and, in fact, in a physical and dynamical problem 
it is the elliptic integral which arises in the course of the 
work ; for instance in the form of the Equation of Energy, 

$(dx/dt) z = X, so that V 2 t=/dx/JX-, 

and now, when X is a cubic or quartic function of x, so that 
d 2 x/dt 2 is a quadratic or cubic, as in 30, the integral is called 
an elliptic integral of the first kind ; and we have to follow 
Abel and determine the elliptic function which expresses x as 
a function of t. 

To accomplish this, it will be useful to employ the notation 
of the inverse functions, given by Clifford (Proc. London 
Math. Society, vol. vii., p. 29 ; Mathematical Papers, p. 207) 
analogous to those used in Trigonometry for the inverse 
circular functions ; and to make a collection of all the important 
cases that can occur. 

34. The Circular and Hyperbolic Integrals. 

Starting with the circular functions, sin x, cos x, tan x, cot x } 

... , we have, in the ordinary notation, 

30 



THE ELLIPTIC INTEGRALS. 31 

Cx fa 

I =sin- 1 ^ = cos- 1 /v /(l-^ 2 X 
*/ ^ \J- ~*~ x j 



r\ r 7~ 

/ aa " =cos- 1 ^ = sin- 1 x /(l-a; 2 ), 

+s ^/^-L ~~ X J 

x 

f 



X 

x dx 1 

^-j =cot-^ = tan--,etc. 

X 

We can employ a similar notation with the hyperbolic func 
tions, cosh x, sinh x, tanh x, coth #, . . . , and write 



X Jrf 

- = 




x-\-\ 

etc.; 



and the analogy with the circular functions is now complete, 
and the results can be more easily remembered and written 
down, than when the logarithmic function alone is employed. 

To avoid complications due to the multiplicity of the 
values of these and subsequent integrals, in consequence of the 
variable x assuming complex values and performing circuits of 
contours round the poles of the integral, we suppose for the 
present that x is real, and increases or diminishes continually, 
so as to assume all real values once only between the limits of 
integration; also that the positive sign is taken with the 
radical under the sign of integration ; we thus obtain what is 
called the principal value of the integral or inverse function. 

35. The Elliptic Integrals. 

With the elliptic functions, snu, cnu, dnw, we have ( 7) 

d SD.U dcnu ddnu 

-j = en u dn u, -. = sn u dn u, j = /c 2 sn u en u : 
du du du 



32 THE ELLIPTIC INTEGRALS. 

and cn 2 i> = 1 sn 2 u, dn z u = 1 /c 2 sn 2 w, ; 

so that, if x = snu, then en u = ^/(l a? 2 ), 



/ ^^ 

7(i-^.i- K vr u==sn " la; or sn 1(x> K) - ....... (1: 

when the modulus K is required to be put in evidence. 

Putting x=l makes the integral equal to K, the quarter 
period corresponding to the modulus K ( 11). 

Similarly, with 

x = en u, then sn u = */( 1 x 2 ), dn u A 



= -sn u nu = - 



/i ^^ 

y (1 -^. K HA*r w = cn la or 



so that the integral is K when the lower limit is 0. 
Again, with 

x = dn u t then >c sn u = ^/(l x 2 ), K en u = ^/(a 2 Ac 2 ) ; 



and ~=- K *suucnu=-J(l-x*.x*-K 2 ) ) 

du 



We may also put x = iuu, using Gudermann s abbreviation 
of tn u for tan am u ; and now 



,_- J. 1 / \ / A \ 

and the integral is K when the upper limit is oo . 

Putting & = sin0, cos 0, A</>, or tan< in (1), (2), (3), or (4), 
reduces the integral to 

/(I /c 2 sin 2 0) ~ -cZ0 = u = F(0, /c) 

o 

so that 

= am u, and cos = en u, sin = sn u, A0 = dn u, tan < = tn u. 



THE ELLIPTIC INTEGRALS. 33 



36. Thus, with a>b>x, 



x dx 1 .fx b 

" (5 



indicating that we must put x b sin ; and then the integral 
is reduced to 



a a V a 

o 



Similarly, with oc > x > a, 

f x dx _1 Ja b\ , 

J (tf-a?.x*-}? a 81 WJ" 



indicating the substitution x a cosec (or a cec 0, as Dr. 
Glaisher writes it). 

Thus, for instance, with oo>o?>l//c, 

f^_ dx VJL \ 

J x/(! ~ ^ 2 1 ~ * 2 z 2 ) ~ W V 
Again, 



; 



37. As numerical examples, 

/i r / r 
-^ 



the integration required in the rectification of the lemniscate 
r >2 = a 2 cos20; so that r = a cn(^/2 s/a, J 



with Dr. Glaisher s notation ( 17) of new for 1/cnu. 

G.E.F. C 



/ 

34 THE ELLIPTIC INTEGRALS. 

Consider also the vibrations given by the dynamical 
equation d 2 x/dt 2 = 2n 2 x(c 2 x 2 ), 

as in 30; so that x = gives the point of stable equilibrium, 
and x = c gives the points of unstable equilibrium. 

Integrating, supposing the motion to start from rest where 
x = b, i (dx/dt) 2 = C- n 2 c 






(i.) When b 2 <c 2 , the motion is at the outset towards the 
origin, and dx/dt = n^/(a 2 x 2 .b 2 x 2 ), 
writing a 2 for 2c 2 b 2 ; so that 

P dx f dx _f x dx 

n Jj(a 2 -x 2 .b 2 -x 2 )-J JXj JX 

x 

= -\K sn~V ), with modulus , by (5) ; 
a\ b/ a J 

or x = bsu(Kant). 

(ii.) When b 2 = c 2 , dx/dt = n(b 2 - x 2 ) ; 
and, by 34, the ultimate state of motion is given by 

x = b tanh bnt, or b coth but, 

according as the motion falls away from the position of 
unstable equilibrium, towards or away from the origin. 
(iii.) When c 2 < 6 2 < 2c 2 , 
dx/dt = 



r* dx r M dx _ 

J x 2 -a 2 .x 2 -b 2 )~J X 



J(x 2 -a 2 .x 2 -b 2 ) 

b b 



(iv.) When b 2 = 2c 2 , 

nt = / ,, 2 pr = ,- sec ~ V> 

6 

or x = b sec bnt. 

(v.) When 6 2 > 2c 2 , we must write a 2 for 6 2 - 2c 2 ; and now 
dx/dt = + nj (a 2 + x 2 .x 2 - b 2 }, 
dx 



_ 1 J6 a 

~v/( 2 + 6 2 ) CI] U /v /(^ 2 + 
or x = 6/cn /v /(a 2 + b 2 )nt =b nc^(a 2 + b 2 )nt. 



THE ELLIPTIC INTEGRALS. 35 

38. So far the function X has been treated as an even 
quartic function of x, or as a quadratic function of a; 2 , resolved 
into two real factors ; but according to Prof. Felix Klein there 
are certain advantages in considering the integrals obtained 
by writing x 2 =z, in (1), (2), (3) ; and then, waiting k for * 2 , 



f 

J 



or 2cn-V(l-s), or 2dn- 1 v /( 1 -^) ............. C 11 ) 

Conversely, by writing for z the values & 2 , 1 # 2 , 1 kx 2 , we 
reproduce the integrals (1), (2), (3) from (11), by the simplest 
quadric transformations; and it will not cause confusion if 
we sometimes call k the modulus. 

For these and various other reasons, Prof. Klein suggests 
(Math. Ann. XIV., p. 116) that we should consider (11) as a 
more canonical form of the elliptic integral than (1), the form 
with which Legendre and Jacobi have worked. 

39. Now, with X = x a.x {3.x y, and a>/3>y, 
we have, if GO > x > a, 



f 
J 



q-y 



, (12) 



indicating that we must put 

x y = (a y)cec 2 0, x a = (a y)cot 2 0, 
and then x - /3 = (/3 - y) A 2 cec 2 0, 

to reduce the integral to Legendre s canonical form 



=(! - k sin 2 
o 



Similarly, by putting x - a = (a - /3) tan 2 0, x - /3 = (a - 



/ x 



a-y.o;-/3 
where Jl/ is used throughout to denote J^/(a y). 



.(13) 



36 THE ELLIPTIC INTEGRALS. 

Thus, with -ao>x>l/k, integral (11) becomes 



f 

j 



9 1 L 



-" SI 



_ 

J(x.l-x.I-kx) 

l/fc 

1 I l-k , . ll-k.x 
- 1 - -=2dn- 1 A/- J~. 
.x 1 v cc 1 



40. When a>#>/3, Z is negative, and 



y 

_ la-.x- 

- 



(14) 



/-J/fe 
J *J(- 



= cn -i /toir:* = dn -i J^=y ...... (15); 

\a jS.aJ y \cc-y 

and now the modulus K is given by K 2 = k f = (a /3)/(a y) t 
and the modulus is therefore complementary to the modulus 
in (12) and (13) ; and the form of the result in these and other 
subsequent integrals indicates the substitution required to 
reduce the integral to Legendre s standard form ( 4) ; while 
the results can be verified by differentiation. 

Thus, with l//c>&>l, integral (11) is imaginary and may 
be written 

dx I -fee 

- 



f* dx _ l I x-l 

JJ(x.l-x.I-kx)~ yi-k.x 

.kfaB-t^^-llfc-^ mod. *T; 
i denoting *J( 1). 



THE ELLIPTIC INTEGRALS. 37 

41. When (3>x> y, X is again positive, and 
P Mdx la-y.fi-x 

7T -^p-y.a-x 

-m^JyptelZ.-to-iJsze. . ..(16) 

V/3-y.a-x \a 05 

/ 3/tZa /a-y 

^ = sn - V^ 

x 

..................... (17) 



with fc = 08-y)/( a -y), as in (12) and (13). 
Thus 
/- 1 do? ll-x 

j _ v or * / _ 

J J(x.l-x.\-kc) \1-A 



while the result is as in (11) when the lower limit is 0. 
42. When y>x> oc,JTis negative, and 



=cn 



x Mdx ja-y 

f. ^r = sn- 1 A /- - L 
J(-X) \a-x 



with modulus k = (a /3)/(a y), as in (14) and (15). 
Thus, with 0>x> oc , integral (11) becomes 

/ J* . 



_ ---- = 2idn~ 1 / */- --- j 



^ o- / ! 

.i-a:.i- = sn Vr- 



a 



38 THE ELLIPTIC INTEGRALS. 

43. We notice that the substitution 



_ 

,r --- , UI 



- ,r --- , ^ 75 ---- , - , 

# y p y & y p y # y a y 
makes 

dx y dy 



x y 

or changes (12) into (17), or (13) into (16). 

Thus 

dy 



r dx . 

J(x-a.x-/3.x- 7 ) 



j(y-a.y-/3.y- 7 ) 

a 7 

where K 2 = k = (/3 y)/(a y). 
Again the substitution 



., or = or = 

a-/3 a-2/ a-/3 a~2/ a-y a-y 
changes (14) into (19), or (15) into (18) ; and shows that 
dx fy dy 2A" 



where k = K 2 = (a- ft) /(a - y). 

The substitution which changes any one integral into another 
is obvious by inspection of the preceding results. 

44. Thus the integral fdxj^/X can be written down, ex 
pressed by inverse elliptic functions, when X is a cubic form 
in x, resolved into its three real linear factors. 

For example, with a 2 > 6 2 > c 2 , 

d\ 2 J c 2 +X 



\ 

an integral occurring in the mathematical theories of Electricity, 
Magnetism, and Hydrodynamics, in connexion with ellipsoids. 

As another example, the student may prove that 



r _dS_ 47ra6c Jc_ /a 2 -6 2 \ 

J (xjaY + (y~lb) z + (z l c)* ~ J(a 2 - c 2 ) C W V a 2 - c 2 7 
when the integration is extended over the surface S of the 
sphere x* + y* + z 2 = r* 

(W. Burnside, Math. Tripos, 1881). 



THE ELLIPTIC INTEGRALS. 39 

45. When two of the roots, /3 and y suppose, of the cubic 
X = are complex, we combine (x ($)(x y) into the real 
quadratic (#-m) 2 + n 2 , suppose ; so that X = x a . (x m) 2 -f n 2 - 
Now we substitute 

X = (a;-m) 2 +7i 2 
y ~(x-af~ x-a 

a quadric substitution, the #rap/i of which is a hyperbola, and 
find the turning values of ?/, say y 1 and ?/ 3 , the values of y 
which make the quadratic in x, 



have equal roots ; so that y : and y 3 are the roots of 

= 0, or ^ + ( m - a )y 



-, - B 

Then y - yi = -, y-?,= . 

(Zy_(a?-a? 1 Xa?-g 8 ). 

anQ 7 \9 5 

aa? (OB a) 2 

x l and 3 denoting the values of x corresponding to y l and 2/3, 
and therefore denoting the roots of the quadratic equation 

x 2 - 2ax 4- 2am - m 2 - n 2 = ; 
so that x = m + x = m 



Then 



/ 

J 



_r ^ 




by (12), with k / = yJ(y l y^), k= 

J \ /* t/ 1/ \t/ 1 t/ o/ 

since 2/ : is positive and y z negative, or y l > y > > y y 
Again, with the same substitution, 

dx r dy 



cn-i /ft-y 

2/3) >2/i-2/ 

-^ rcn- 1 ^^ ...... (23) 





by (19), to a modulus k the complementary modulus of (22), 
namely k = yj(y^ - y s ). 



40 THE ELLIPTIC INTEGRALS. 

46. We denote (a-m) z + n* by H\ and then 



and by means of the same substitution as in 45, 
dx 



en 



, (24); 

dx I JH--(a-x) ,\ 

~v# CI iff+to^sy*) 



r , (25); 

indicating that the substitutions x a or a x = H(^ 
reduce the integrals to Legendre s standard form ; also that 

Thus, as numerical examples, 

/CO fJ ] / "I /O \ 

tit*/ JL - 1 / ^ ^ ^v \ 






i 



/" g da; J_ 

y x/a-^ 3 )"^ 



with 2/c/ = J = sin 30, K = sin 15, K = sin 75. 

47. We notice that ^ = ITT when x = aH ) so that 

" 



/* ^ 

y /v /{ a; --(-W ) 2 +^ 2 } 

+H 

= /: a+ ^L _J?L_ 
Jj{x-a.(x mf+n? 



a x . (x m) 2 + n 2 } 

(27) 



-00 



/OO xV/v, 
3 ,,= 
/ 1 / 7**^ 1 \ 

\/3+l 



ix r^^ dx 



THE ELLIPTIC INTEGRALS 

dx 



/* dx r 

x/a-^)y 



- -v/3+l 

But, by the Cubic substitution x = (4< 



so that 



or ^(sin 75) = ^/3-F(sin 15), 

that is, K IK= ^/3, if /c = sin 15, as stated in 12. 

48. Degenerate Elliptic Integrals. 

When the middle root /3 of th,e cubic X = approaches to 
coincidence with either of the extreme roots, a or y, or when 
the pair of imaginary roots become equal, the elliptic integrals 
degenerate into circular or hyperbolic integrals. j> 

We notice, from 16, that when & = 0, sn -1 ^ becomes sin" 1 ^, 
cn 1 ^ becomes cos" 1 ^, etc.; and that, when k = l,sn.- l x becomes 
tanh 1 ^, cn 1 ^ or dn- 1 ^ becomes sech" 1 ^, and tn- 1 ^ becomes 
sinh ~ l x. 

Thus, when k = l, the integral (11) 

/dx _ _ r dx 
/(x.l-x. i-kx) J(l-x)Jx 

= 2 tanh- = 2 sech~ 



__. 
1 a; \1-OJ a? 

This supposes that x < 1 ; but with oo > x > 1, 

^^ oo 7 

y (!)> = 2 coth " V^ = ^ cosech- V(s- 1) 

= 2 sinh- 1 J - = 2 cosh-i/ = sinh -i V^. 
\ic-l \-l ic-1 

But when & = 0, the integral (11) becomes 

. 

= 2sm~ 1 /v /ic 
- 2 cos- 



r 1 dx _ _j 

x = 2sin- 1 v /(l X) = TT sm~ l 2^(x. Ix). 



42 THE ELLIPTIC INTEGRALS. 

49. Making /3 = y, or a, in the integrals (12) to (19), and 
still denoting J*/(a y) by M, then 
(i.) with oo>aj>a, 

Mdx Ix-a la y 

/ =Sin~ 1 A / --- ^ 
VOJ- VoJ 



= COS 



- 



y 

1 i*/(a y & a) 

- 1 v 



-flfcfcc . ! a-a 

sm = cos 



, /a-y 
" 1 ,*/ -- ^-; 

A aj-y 

T 60 Mdx , . /a-y . , , /a-y 

/7 -- - -- ^ = - --- ^- = - 1 -- * 

J (x-a 



this integral being infinite when a; = a. 
(ii.) With a > a; > y, 

f* Mdx . Ix-y /a-y 

/7 -- ~ -- - --- ^- -- * 

y(a- 



7 



which is infinite when x = a ; 

f a Mdx . la-x 

/7~ -- \~~77 - r = smh~ 1 A /- -^ 
J (x-y)J(a-x) \iC- 



which is infinite when a? = y. 
(iii.) With y > cc > oo , 

Mdx . , /y-aj /-y 

~ ~ = 1 - ---- = " 1 -- L 

a? 



a-x 



, = cosh - J A / = sinh - \/ 
"a an Vy a? \v x 



this last integral being infinite when cc = y. 

The limits have been chosen so as to exclude these infinite 
values. 

50. Weierstrass s Elliptic Functions defined. 

When the general cubic expression X is given, not resolved 
into factors, then Weierstrass s notation becomes useful, and 
may be defined here. 



THE ELLIPTIC INTEGRALS. 43 

Weierstrass writes s+f for x, and chooses / so as to make 
s 2 disappear in the new value of X, which he denotes by J5 ; 
and thus S = 4s 3 g 2 s <? 3 , 

where g 2 and g 3 are called the invariants ; so that the integral 
dx ds x ds 

u suppose 



f 

J 



and now, inverting the function in Abel s manner, s is an 
elliptic function of u, denoted by pit- in Weierstrass s notation, 
so that 



=rl8i or p-(; g v g s ) ............. (A) 

when the invariants g 2 and (/ 3 are to be put in evidence. 

51. In Weierstrass s notation we are independent of the 
particular resolution of S into factors ; but by what precedes 
in equation (12), if, when $ is resolved into real factors, 

S = 4(s 6^(8 e. 2 )(s e B ), with e l > e. 2 > e 3 , 
then, with x > 6> > e v 

C ds 1 _^ _ l /e 1 ~<? 3 

e)^ V - 



s e l 1 , , !* e< 

= vdn~ 



by (12); so that 



(B) 



The value of u for s = e l is denoted by o^, and called the 
real half period; and by (20) we notice that 

Wl =f x ^=r^ s =^--- (28) 

e x ^ e 3 

andby(13)and(B),/ %< -^ < ==p- 1 ( ei ~ gg ei ~ g8 + g 1 ) (29) 

w/ v ^ ^ s e i 

With e 2 >s>e 3 , ^jS is again real, and by (16), (17), and (B), 



44 THE ELLIPTIC INTEGRALS. 

52. For values of s between e 1 and e 2 , or between e z and 
~ , *J& is imaginary ; however, the value of Jdsj \JS be 
tween the limits e z and oo is denoted by o> 3 , and called the 
imaginary half period ; so that, by (21), 

/-i ds r<* ds _ iK f 

^ = J jsy X/S T^S) ............. (82) 

2 

and, from (12) and (14), 

* 2 = ( 2 - 8 )/(l ~ *3)> K* = ( e i ~ 2 )/(l ~ S )- 

Also, from (14) and (15), with e 1 > s> e 2 , 

/ c i ds . 1/01 e^e, o \ 

^s =tp ( - t-s <i; ^ ~4 ......... (33) 



/ s <s . ,/e, e 2 .e 2 60 \ 

^=^-v ;_; -*-<*> * -*)j ....... 



and, from (18) and (19), with e s > s > oo , 

/ e a ds 



/S fj Q 
-Ta^P H-*; 9 -9 3 ) (36) 
V^ 



53. The quantity gf 2 7# 3 2 is called the discriminant, and 
is denoted by A ; it is called the discriminant, because the 
roots of 8 = are all three real, or one real and two imaginary, 
according as A is positive or negative ; and A = 0, when two 
roots are equal. 

Since S = 4 3 - g 2 s -g 3 = 4(s - e^s - e 2 )(s - e^\ 

therefore e^e^e^O, 

and g 2 = - 4(e 2 e s + e^ + e^ = 2(e* + e 2 2 + 



Therefore 



and 



This quantity 5f 2 3 /A is called by Klein the absolute invariant, 
and denoted by J ; and then, with k for /c 2 , 



- ,_ _- r 

~ 2 2 



THE ELLIPTIC INTEGRALS. 45 

54. For the present we reserve the difficulties of interpreta 
tion of the multiple values of the integral u =fds/*JS, due to s 
being allowed to assume complex values, and to perform 
circuits round the poles, branch points, or critical points, so 
called, of the integral, given by the roots of S = 0. 

We suppose the variable s to pass once through all real 
values from oo to oo ; and now 

(i.) oo > s > e, 



/ 



or u=v l -dslJS=< a ,- 9 --*-*+e, ........ (37) 



which, employing the direct functions, expresses the relation 

*->-*-*=$? ................... (38> 

(ii.) e l >s>e 2 , 



(39) 

or u = CD, + a), 



-e 2 ; g r -g s ......... (40) 

(iii.) e 2 > s> e 3 , 
u = w + w 



(41) 

or u = 20)! -f- c 3 /ds/^/S 
< 3 

(iv.) e 3 > s > co , 



-jr.); (43) 

/ 



46 THE ELLIPTIC INTEGRALS. 

or u = 2^! + 2ft> 3 /ds/^/8 

00 

Thus /ds/^S=2a) l + 2co 3 , (45) 



and 2ft)! is called the real period, and 2w 3 the imaginary 
period of Weierstrass s elliptic function pu. 

With Argand s geometrical representation of a complex 
quantity, such as x + iy, the complex quantity 

u = ta> l + 1f( l 6 3 (0<t<l,0<t <l) 

represents all points lying inside a rectangle, called the period 
parallelogram. 

As 8 or $u diminishes continually from oo to GO , the argu 
ment u describes the contour of this rectangle ; and for 



(iii.) t^ + a> 3 (I>t> 0), (iv.) t a> s (l>if> 0), 
the values of s or $>u- are real, and lie in the intervals 

(i.) ao>s>e lt (ii.) e 1 >s>e 2 , (iii.) e 2 >s>e 3 , (iv.) e 3 >s> CXD ; 
while the corresponding values of $ u are taken as 

(i.) negative, (ii.) positive imaginary, 

(iii.) positive, (iv.) negative imaginary. 

For any point u inside the rectangle $u assumes a complex 
value. (Schwarz, Elliptische Functionen, p. 74.) 

55. In the same way, with the integral (11), denoting its 
value between the limits oo and z by u, 
(i.) oo >z>l/& (39), 

= 2K-2sn-^f^\ ................... (46) 

(ii.) l/k > z > 1 ( 40), 



(47) 
(iii.) 1>0>0(41), 



- 

kz 

.................................. ( 48 ) 



THE ELLIPTIC INTEGRALS. 47 

(iv.) 0>2?>-oc (42), 



(49) 
Therefore f~ r =-^- 7 ^ = 4A r +4iJT, .............. (50) 

^/ *J(Z . 1 2? . 1 A 1 ) 

and 4JT and 4^Ar are called the real and imaginary periods of 
the corresponding elliptic function, in this case sn 2 |i,. 

56. But if we take Legendre s and Jacobi s fundamental 
integral Jdxj ^/X, where X = 1 x 2 . 1 K 2 x 2 , and denote 
r 
Idxj ^/X by u, then, by the preceding article, with x~ for 0, 

(i.) CC>X>\JK, 

1 / ^2^2 _1 

" KV& ~~ -L / K -i \ 

(ol) 



(ii.) I 

" I - 



/C 

-, K ^ ........................ (52) 

(iii.) 1>*> 1, 



= 



J- /C 

(iv.) -1> a; >-!/*, 



/c 
2i^-^sn- 1 / - 1 , K .................... (54) 

(v.) -l//t 



..................................... (55) 



Therefore *,(I-.x*.I-K*a?)-*dx = 4K+2iK -, .............. (56) 



-00 



and 4/f and 2iK are called the periods of the elliptic func 
tion sn u. 



48 THE ELLIPTIC INTEGRALS. 

57. If, with l>x>-l, and X=I-x 2 .I-A z , we denote 

r* r i 

the iategr&lfdx/JX by u ; then/dfe/^/Z=JT ( 11); and ( 41) 



or, employing the direct functions, 

/ T 

sn 

and then ( 17) 



\ 1X 2 

- w ) == or cdw; ......... (57) 



(58) 
< 59 > 



relations analogous to equation (38) ; or to the relations 

sin( JTT 0) = cos 0, cos( JTT 6) = sin 9, 
of the circular functions of Trigonometry. 

58. When the discriminant A of 8 is negative, and two of 
the roots of the equation 8 = are imaginary, we take e 2 as 
the real root, and combine the product s e^.s e^ into 
(s m) 2 +7i 2 , as in 45 ; and since 



therefore m = Je 2 , # 2 = 3e 2 2 47i 2 , # 3 = 
while H 2 = (6 2 - m) 2 + ?i 2 = f e* + ?^ 2 , 

4 A 2 = n*/H 2 = 47i 2 /(9^ 2 2 
1 - 16/cV 2 = 3^ 2 /(9e 2 2 + 4 2 ), 
A = 3 



, fha r_ 2 _ _ 

= " 



_ 
108A 2 

- 2 



59. Now, as in 45, by means of the quadric substitution, 

,2 
,....-. (60) 



THE ELLIPTIC INTEGRALS. 49 

da- (s-e,) 2 -# 2 (s-s)( s -s) 
ds= (s-e z f (A) 2 

(s-^) 2 (s-s 3 ) 2 

while oe^-, o--^ = _-__, 

provided S 1 = 



Thence s l + s 3 = 2e 2 = J( l + e 3 ) - e 2 - e 2 = - f 6 2 ~ e z 5 
or 6 2 = | e 2 ; on the supposition that e l + e 2 + e 3 = ; 
and e-j = e 2 + 2H, e 2 = 2e 2 , e 3 = e 2 2^. 

s -T (s-ejda- 



fA s -T (s-e 

J JSJ^-^- 



ei.or-e^o e s ) 



cr 



where 2 = 4(<r e 1 )(cr e 2 )(o- e 3 ) = 4<7 3 y 2 o- y 3 , 
suppose ; and the discriminant A 7 of 2 is now positive. 



60. Now, y 9 =-4(e 2 
y 3 = 46^3 
A 7 = y 2 3 - 27y 3 2 = 256^ 2 (4JJ 2 - 9<? 2 2 ) 2 . 

.,, , e. 7 -e 3 2#-3e 2 , 6l -e 9 

Also with A 2 = * -3=a 2 , X 2 =^ -- ^ = 

e l~ 6 3 *" 6 1~ C 3 

4ff ! _9^_^ 2 3 

~H 2 lA ~ 



Denoting by J the absolute invariant of 2, then ( 53) 
y 2 3 = 4 (1-X 2 X /2 ) 3 
A 27 X 4 X /4 
If we put 4X 2 \ /2 = 1/T, then 
(4 T / -1) 3 r 

27T d 
while, with 4/cV 2 = r in (D), 

r_- 8 



Now, if 2/c/c = 2XX , then T T /= 1, the relation which holds in 
the transformation from a negative discriminant in $ to a 
positive discriminant in 2. 

If we equate the values of J in (C) and (E), we find 
(1-fc) 2 & 1 



G.E.F. 



50 THE ELLIPTIC INTEGRALS. 

61. When A is negative, and when we know the real factor 

s <? 2 of $; so that, with Je 2 2 + ri 2 = 



then, with # 2 = J(9e 2 2 + 4ii 2 ), and expressed as in 46, 
f ds I 8 -e 2 -H 

U=/ -7-3 = /rrCn- 1 yv, .............. (62) 

y x/>3 2*/-ti s e 2 + H 
with 2KK=n/H ; so that 

-- e 2 +H-(e 2 -H)cn(2u*/H) . 

, or p*= _J__ 2 ; (63) 



by means of which we change from Weierstrass s notation to 
Jacobi s and vice versa, when A is negative. 

Thus, for example, if # 2 = 0, then <? 2 = (J<7 3 )*, n 2 = f e 2 2 , # 2 = 3g 2 2 ; 
and, as in 46, 



3 + l)ft.93) i Sinl5 } 
-lXi^)* / 

/ ^g 
(4iM7s) = rl(8; -^ 

--^ sin 75-1- 
i S 



62. Supposing s to range from oo to oo in the integral 
r> 

= /ds/^/S, when A is negative, then 

* 

(i.) <x> 



where o> 2 denotes /du/tJS, the real half period of n. 
(ii.) e 2 >s>-cc, 



where w/ denotes /ds/^S, a pure imaginary quantity, called 

the imaginary half period of $m ; and the period parallelogram 
( 55) is now bounded by o> 2 and , , as adjacent sides. 
Also (47), mrKIJ-H^-iriJH. ,.-(66) 



THE ELLIPTIC INTEGRALS. 51 

63. Treating in the same way the integral (2), 
_ /"" dx 

I /(~\ _ /r 2 *- 2 4- i^rV 
*/ /^/V- 1 - * K ** ^ 

by replacing z by 1 z 2 in 38, 55 ; 
(i.) oo>a>l, 

(ii.) 



(68) 

(iii.) 1 > x> co , 

u= iK + 2K+icn-\-l/x, K) 

.......... (69) 



64. By the substitution x 2 = l/y, the integral 
dx " dy 



i r-ds 7 

~^AJ 7S " 



on putting y = s \B/A ; which can be expressed by Weier- 
strass notation, or by the notation of Jacobi, when the factors 
of the denominator are known, as in equations (12) to (19) ; 

f_ E+Fx _, 

J J(A + Ba? + Cx* + Dx Q ) 

can thus be reduced to elliptic integrals, of the form considered 
in 39-61, the first term by the substitution x* = I/y, and the 
second term by the substitution x- = z. 

rnu a a*dr a 

rhus 



the integration required in the rectification of 7 13 = a 3 cos W. 
But by substituting ? >2 /a 2 = l/y, we find 
a s dr r ady 



s = 

o 


so that 



52 THE ELLIPTIC INTEGRALS. 

65. Write X for ce 2 - a 2 . x 2 - b 2 . x 2 - c 2 , where a 2 >6 2 >c 2 ; 
and write Jf for b^/(a 2 c 2 ) ; then we find, on substituting 
y for I/O) 2 , and taking a, /3, y for 1/c 2 , 1/6 2 , I/a 2 ; 
(i.) oo > x 2 > a 2 , comparing with equation (18), 

fMdx_ l la- 2 -x~ 2 _ Ib 2 .x 2 -a 2 

J -jx~ VF*=~ar 2 ~ ^a 2 .x*-b 2 



Ia 2 -b 2 .x 2 Ia 2 -b 2 .x 2 -c 2 

= cn " ~ 1 - 



to modulus 



(ii.) a 2 > a; 2 > 6 2 , comparing with (17) and (16), 
Mdx x /6 2 .a 2 -^ 2 

T" Va 2 -6 2 .o) 2 



~ 



to modulus 



(iii.) b 2 >x 2 > c 2 , on comparison with (15) and (14), 
b Mdx 



f 
J 



" 



~ 



/-* 
J 

** *.a?-x 2 7 - 

2 .......... (75) 



to modulus 



(iv.) c 2 > a 2 > 0, on comparison with (13) and (12), 

f Mdx 1 lb*.c*-x z 

J J(-X)- Vc 2 .6 2 -a5 2 



THE ELLIPTIC INTEGRALS. 53 

f x Mdx 1 Ia z -c 2 .x* 

J /(-*)" Vc 2 .a 2 -a 2 

/a 2 .c 2 -o; 2 Ia?.b 2 -x* 

= cn- 1 A /^ ^ H = dn- 1 /x /r; ? 5 , (77) 

\c 2 .a 2 a? 2 yb 2 .a? x 2 

2 *- 2 
to modulus 



66. When X is a quartic function of x, and we know a factor, 
x a, of X, then the substitution x a = l/y reduces 

JclxJ^/X to the form M/dy/^/Y, 

where Fis a cubic function of y\ and this form can be treated 
by the preceding rules. 

But, independently, if we can resolve X into four real linear 
factors, x a, x /3. x y, x S, 

so that X = x a . x /3 . x y . x 3, 

and we suppose that a > /3 > y > S ; then with 

(i.) oo > x > a, 
dx 






. . ( 78) 



indicating that we must put 

. 9 /3 S.X a 9j 
sm 2 = " 5 - 5 , cos 2 = 




9 S.X a 9j a /3.x 

2 " - 2 - - 



to reduce the integral to the standard form ( 4) 
2 



and then 



a y .p o 

the anharmonic ratio of the four points A,B,C, D, the >cte of 
the integral ( 54), given by x = a, /8 y, 5. 

The verification by differentiation is a useful exercise for the 
student. 



54 THE ELLIPTIC INTEGRALS. 

(ii.) With a > x > /3, we change the sign of X to make the 
integral real; and now, writing M for J^/(ct-y./3-<S) throughout, 
r a Mdx 

-i /^L^- i /q-<?-a-ft_ i i la-S.x-y 
^a-$.x-S~ y a -/3.x-S~ Va-y.a-<T" 

r*Mda 

I 8n -^l^^^ 

but now the modulus K is the complementary modulus to /c, so 

j. i j /o 7 / ct ~~ o . *y ~~ o 

tnat K K = ^ ; 

a y . p ^ 

the different forms of the result indicate the appropriate substi 
tution required for reducing the integral to the Legendrian form. 

(iii.) With /3 > x > y, X is again positive, and 



/ 

X 

l la-y./3-X . la-6.X-y , . la-6.X-S /01 . 

= sn- 1 A /-^- ! ^ = cn- 1 A /-> r J - J: =dn- 1 A/ )0 % ,....(81) 
\/3-y.a- ^l/3-y.a-x Vp-3.a-X 

/* Mdx 

^ X 

l/3-S.x-y ly-S./3-X ly-S.a-X 

-sn-M^ -isBcn-Mi -^ = dn- 1 -^- - 
V p-y.X-o Vp-y.aJ-o, 

with the same modulus AC as in (78). 
(iv.) With y>x> S, X is negative, and 
Mdx 



-S.8-x \a-y.8-x v 



y-S./3x yyS./3x \o-y.p- 

x dx 



-i la-y.x-S_ _ 1 la-S.y-x_ 1 ^_ 1 ja-S./S-x 
with the modulus of (79) and (80). 



la-S . /3-x , . 
A / j- ,....(84) 
\y-o.a-a; 



THE ELLIPTIC INTEGRALS. 55 

(v.) With S>x> x , X is positive, and 

f 7^ 

"~4^ :: -~ (85) 



with the original modulus of (78), (81), and (82). 

67. Landens Transformation. 

When Legendre s arid Jacobi s standard integral (1) is 
treated as a particular case of these integrals (81) and (82), we 
write a = l/X, = l,y= -!,<$= -1/X, so that Jlf=J(l+X)/X; 
and now, with y for variable, 



/l+X.1-3/ _ x /l-X.l + y_ /1-X.l + Xy , . 

V~2J^X - V 2.1-X - Vl+X.l-X-" (8 



where the modulus AT is now given by /r = 4X/(l + X) 2 , so that 



and we are thus introduced to Landens transformation, to be 
discussed hereafter. 

Changing, in 41, x into y 2 , and k into X-, we find 
dy 



with modulus X ; indicating, on comparison with (86), results 
such as 



which can be translated into the various forms of Landen s 
quadric transformation. 



56 THE ELLIPTIC INTEGRALS. 

Denoting integrals (86) and (88) by u and v, then 



ojj. y<y, i\i _ ^, 

i-xy 

cn 2 (v, X)=-jJ^y4, dn 2 (v, \) ss ~^- t (91) 

whence sn(v, X) = ( 1 + g/ M^>^)cnK ^ etc (92 ^ 

We can easily prove, or verify by differentiation, that 



= _ - _ 

-- 2 ^ 



to the same modulus /c = 2 x /X/(l+X); so that, denoting this 
integral by u, and denoting sn(w, K) by x, then 



- y 



or dn(w> K ) = , ; nd(M;/c) = .-. f 

-- 1 A 



since ^/ sn (^ ^) where v = (1 +O M 5 
and thence 

X)nd(^,jc) J ............... (96) 

^) n d(^/c); ............... (97) 

(Cayley, Elliptic Functions, p. 183). 
The relation (92) between x and ^/, namely, 






thus leads to the differential relation 

dx 



(98) 



THE ELLIPTIC INTEGRALS. 57 

68. The six anharmonic ratios of a, /5, y, S, arising by per 
mutation or substitution, give rise to six values of the modulus 
k, given by 

^i-^ rV-iL^i ................... (99) 

or sin 2 #, cec 2 #, cos 2 #, sec 2 #, -cot 2 $, -tan 2 #, iffc = sin 2 0; 
or tanh%, coth%, sech 2 u, coslrw, - cech 2 u, - sinh 2 u, if k = tanh 2 u. 
We may notice that the expression for / in (D) of 53 is 
unaltered if for k we substitute any of these other five values ; 
and, on comparison with Weierstrass s notation, 

J 
so that we may put 



- nom 

~ A = 25(5 -" (1 

and then ^ = T V(2-Q, 6 2 = T V(-l + 2fc), e 3 = r V(-l-&) ; 
so that h = (e. 2 e^l(e l e^) ) as in 51. 

69. Degenerate Forms of the Elliptic Integral. 

When two of the roots a, /3, y, S become equal, the corre 
sponding integrals degenerate into circular and hyperbolic 
integrals, which can easily be written down, on noticing as 
before ( 48) that (i.) when k = 0, STL~ I X becomes sin" 1 ^, cn^se 
becomes cos" 1 ^, etc; (ii.) when k=l, sn 1 ^ becomes tanh" 1 ^, 
cn" 1 ^ or dn -1 ^ becomes sech^, and tn~ 1 o? becomes sinh" 1 ^. 

When two of them are equal, we may replace the four, 
quantities a, ft y, S by the three distinct quantities a, 6, c, 
suppose, where a > b > c ; and now the degenerate elliptic 
integrals fall into three classes, I., II., III. 

I. Writing M for J^/(a b . a c) ; then 

(i.) oc > x > a, 
f Mdx . , , la b.x c , , la c.x b 

IT - r - 77 -- j - r = 81DJl- 1 A/T - ~ = COSh ~ 1 /v / j - . 

J (x a)tj(x b,x c) \b-c.x a \lb-c.x a 

(ii.) a>x>b, 

f x Mdx , la b.x c . , , la c.x b 

/- - - 77 - T - r = cosh~ 1 /v / I - = sinh~ 1 /v / I - . 
J (a x)fj(x b.x c) yb c.a x yb c.ax 

b 

(iii.) b > x > c, 

Mdx _ _ _ 1 la b.x c_ . _ 1 la c.b x 
(b-x.x-c)~ Vb-c.a-x- ~ ^Jb-c.a-x 



58 THE ELLIPTIC INTEGRALS. 

c.b 



r x Mdx . 1 la b.x c 

// - \ /,-L \ = sin ~ \ rr~ = cos 

J (a x)*J(b x.x c) ^b c.a x 

c 

(iv.) c>x> oo , 

C c Mdx . , la b.c x la c.b x 

/7 -- \ 77i r = sinn~ l \hr- - = cosh~ l *- 1 . 

J (a x)^/(b x:c x) \b c.a x yb c.a x 

X 

II. Writing M for ^(a-b.b-c); then 
(i.) x > x > a, 

C x Mdx . , Ib c.x a la b.x c 

h - IA // v= sin \\ 7 = cos" 1 ,*/ 7. 

J (x b)^/(x a.x c) ya c.x b \a c.x b 

a 

(ii.) a>x>b, 

f a Mdx . Ib c.a x , . la b.x c 

If - 7\ // x = smh~ 1 A / T = cosn~ 1 A/ r. 

J \ x ~ b)*J(a x.x c) ^la c.x b \ja-c. x b 

X 

(iii.) b > x > c, 

f x Mdx , , Ib c.a x la b.x c 

/7I - \~~77 - r = cosn- 1 A/ - -. - = smh~ 1 A / - j - . 

J (b x)+/(a x.x c) \a c.b x \a c.b x 

c 

(iv.) c >x> oo, 

/" c Mdx , Ib c.a x la b.cx 

/7L - T~77 \ = COS" 1 ^/ - j - = ffln Vl - 1 - . 

J (o x)^/(a x.c x) ^a c.b x \a c.b x 



III. Writing . for %J(a-c. b-c) ; then 
(i.) oo > a; > a, 

. la c.x b . , , Ib c.x a 
- 1 - = - 1 -- 



r r . - r - A - j A -- r . 

c)^/(x a.x b) ya b.x c \a b.x c 

(ii.) a>x>b, 

a Mdx la c.x b . , Ib c.a x 

-- r~77 - f^ = cos- 1 ^/ - T - =sin~ 1 /v / - ^ - ; 

x c)^(a x.x b) \a b.x c y a b.x c 

x Mdx _ _ . _j la c.x b_ _ a Ib c.a x 

-c)J(a-x.x-b)~ <\/a-b.x-c~ ~ ^a-b.x-c 



b 

(iii.) b >x> c, 



f b Mdx . la c.b x Ib c.a x 

77 - \ 77 - L - r = Sinh~ 1 A / -- r - = COsh~ 1 A / --- T - . 

J(x-c} t j(a-x.b-x} ^Ja-b.x-c ^a-b.x-c 

X 

(iv.) c > x > oo , 

r Mdx , . la c.b x . , , Ib c.ax 

i -- - - = cosh" 1 A / - j = smh" 1 A/ -- T -- . 

J (c x)*J(a x.b x) ya b.c x ya b.c x 



THE ELLIPTIC INTEGRALS. 



70. When all four roots of the quartic X = are imaginary, 
so that 

(x - a)(x -/3) = (x- m) 2 + n z , (x - -y)(x -S) = (x- 



is reduced by the substitution 



Let us suppose that X is resolved into two quadratic factors, 
so that X is of the form 

X = (ax 2 + 2bx + c)(Ax 2 + 2Bx + C), 

where, by supposition, ac b 2 and ACB 2 are negative, so 
that the roots of X = Q are all imaginary. 



then the maximum and minimum of y, the tui-ning points of 
y, being denoted by y l and y.-,, 



X-L and x. 2 denoting the values of x cormsponding to y l and 
of y ; and now 

dy _2(Ab- aB)( Xl - x)(x - x 9 ) 
dx (Ax 2 + 2Bx + C) 2 

For x is given in terms of y by the solution of 

(Ay-a)x- + 2(By-b)x + Cy-c = 0, ............ (104) 

and this equation has equal roots at the turning points of y, 
which are therefore given by the quadratic equation 



or (AC-&)tf*-(Ac+aC-2Bb)y+ac-&=O t ...... (105) 

and then 

B-b ax + b._ bx+c 

-a y y ~Ax+B~Bx+CT 






Ddy 

dy 



2(Ab-aB) 
and (Ay 1 -a)(a-Ay. z )= - 



AC-B 2 



60 THE ELLIPTIC INTEGRALS. 

sothat 



, . -, , ,,-x f 
which, by (lo), gives / 

* -i /=1L -}- cn-\ /^-4-dn-i / 
V 2/i -2/2 x/2/i v 2/i -2/2 N/2/i >2/i 



with * 2 = 1 - y 2 /y v K " 2 = 

the last expression, by the inverse dn function, being the 
simplest, as expressing a function of an argument oscillating 
between two positive limits, y^ and y 2 . 
71. For example, if 

X = x 4 + 2a 2 o; 2 cos 2a + a 4 

= (x 2 + 2ax sin a + a?)(x 2 - lax sin a + a 2 ), 
and if y (x 2 + 2a& sin a + a 2 )/(a? 2 2acc sin a + a 2 ), 

then #! = , 2/i = tan 2 (j7r + Ja) ; X 2 = a, 2/ 2 = tan 2 (j7r Ja); 
so that // = tan 2 (^7T Ja) = (1 sin a)/(l + sin a) ; 

r _ dx 

J V(^ + ^a^c^cos 2a + a 4 ) 
, 



a 2 (l+sina) 

But, by substituting ~2 = ^j -- > 

ct I 



r 

J 



.. . x , /-, A r\\ 

5 -en- 1 (3; sma) = 7r-cn- 1 - T - -5, ...... (109) 

2a 2a 2 2 



by (2), a reduction of the elliptic integral to a different 
modulus, the modular angle being now a ; affording another 
illustration of Landen s transformation of 07. 
Thus, with a = JTT, equation (108) gives 



where K = (J2-I) 2 (when K \K=\} ; and by (109), 



dx ,l x 2 

etc - 





For other numerical examples, the student may take 

; 2 + 3, etc. 



THE ELLIPTIC INTEGRALS. 61 

72. When two roots only of the quartic X = are imaginary, 
we may still make use of the substitution ( 70) 

y = N/D, where X = ND\ 
but now take ac b 2 negative, and ACB 2 positive. 

Proceeding as before we find that the maximum y l is positive, 
but the minimum y 3 is negative ; and y oscillates between 
and y l for real values of ^/X ; and 

/dx _ 1 

7? ~ J(AC-B*) 

so that, by (14), 



i, ...(110) 
~2/3 

with K 2 = ^(^ - 2/3). * /2 = - 2/&%i - 2/s). 

73. By another method of reduction we shall find 

(Enneper, Elliptische Functionen, p. 23) 
dx 




x - a) 1 ^ 



/j^ _ ^ 

J \/{a-x.x-/3.(x- 



_ lZ(a-x)-H(x-/3) , 



etc. ; where R 2 =( a - m) 2 + n 2 , K 2 =(/3- m) 2 + n z ; 
and Ac 2 = i-a-3 2 -^ 2 - 



so that ZKK =n(a- 13) /H K. 

Degenerate forms occur when a and /3 are equal ; and now 
dx 



n(x-a) 
dx 



THE ELLIPTIC INTEGRALS. 
74. Replacing y by N/D in equations (102), then 



N- Dy. 2 = (a- Ay.^)(x - x 2 ) 2 ; 

so that we may write, according to Mr. R. Russell, 
D = Ax 2 + 2Bx + C = P(x^ - x) 2 + Q(x - x 2 ) 2 , 
N=ax 2 +2bx+c=p(x 1 -x) 2 +q(x-x 2 ) 2 ; .......... (113) 

where P = (Ay l -a)/(y 1 -y.J, Q = (a-Ay 2 )/(y l -y 2 ); 

and p=Py 2 , q = Qy r 

Interesting numerical examples can be constructed by giving 
arbitrary integral values to x v x. 2 , P, Q, p, q ; and now the 

substitution ^ = ^H^2 



, 
fda _C 

J sJX J 



will make, as in 37, 

^ 
J(p+q!?.P+Qz*r 

75. When the factors of the quartic X are unknown, we 
employ Weierstrass s function, and we shall show subsequently 
in Chap. IV. that the elliptic integral Jdx/^/X is reduced to 
Weierstrass s canonical form \Jdsj ^S ( 50) by the substitution 

s=-H/X, 

H denoting the Hessian of the quartic X (Cayley, Elliptic 
Functions, p. 346) ; we may thus write 
~dx T ,/ H 
= 



where g 2 , g 3 are the quadrinvariant and cubinvariant of the 
quartic X or aa) 4 + 46x 3 + 6c 

so that g 2 = ae ^bd + 3c 2 , 

g B = ace + 2bcd ad 2 eb 2 c 3 , 

H = (ac - 6 2 X + 2(ad - bc)x 3 +( 



and the general reduction of the elliptic integral of the first 
kind Jllxj^/X, where X is a cubic or quartic function of x, 
is now complete. 

The application of this general method to the particular 
cases already discussed is left as an exercise for the student. 

76. Systematic Tables of the integrals of the elliptic functions 
sn u, en u, dn u t ns u, ds u, cs u, dc u, nc u, sc u, cd u, sd u, nd u, 
and of their powers have been given by Glaisher (Messenger of 
Mathematics, 1881 ). 



THE ELLIPTIC INTEGRALS. 63 



Suppose yen udu is required ; we may write it 

o 
cnudnudu r cZsnu 1 . 1 

=/ 771 - 9 ^-^ = -sm- 
dnu yV(l JK%n%) /c 



o 
etc. ; so that 



IK. en -M/cZu = cos " x (dn u) = sin ~ 1 (/c sn u) = tan ~ 1 (/c sn u/dn u) 

o 

= J sin " I (%K sn t& dn u) = am(/ci6, l//c), etc. 



Similarly, 

K 



u + KCiiu , dnu+/ccnu , K 

=log -i =logj , etc., 

K dnu /ccnu 



while /dn udu = cos " 1 (cn u) = sin ~ 1 (sn u) = am ^ .......... (1 1 6) 

o 
As an exercise the student may integrate nsu, dsu, ...; also 

sn 3 i6, cn%, dn 3 u, ...; and obtain formulas of reduction for the 
integrals of (sn u) n , (en u) n , (dn u) n , 

As a general method, for (snu} n for instance, we put 
sn 2 u = s ; and now 



s.i-k S r Un suppose - 

By means of the well known formula of reduction, 



for v p =Jx p dxj^/N, where N=ax* + 2 

we have, on comparison, 

a = fc, 6= -1(1 + ^,6=1,^ = 1(71-1); 
so that v p = 2u n , v p+l = 2u n+ to v p - 1 = 2u n - 2 , and 
(n + l)fcie n+2 ?i(l + ^)u n + (?i l)u n - o = sn n ~ l u en u dnu,...(l 1 7) 
the formula of reduction for u n =y(sn u) n du. 

When the limits are and K, we obtain the recurring formula 

(?i+i)^ /l+2 -?i(i+^K+(^-iK- 2 =o, ...... (118) 

ri-n- 

analogous to Wallis s formulas for /(sin or cos 0) n dO. 

o 
The same formulas hold for u n = (cd u) n du, since ( 57) 

cdu = sn(^ u). 

Thus u n is made to depend ultimately on u v already deter 
mined, or on u. 2 ; and a similar procedure will hold for the 
integrals of (en u) n or (sd u) n , (dn -u-) n or (nd u) n , etc. 



(54 THE ELLIPTIC INTEGRALS. 

77. The Elliptic Integral of the Second Kind. 

We may mention here incidentally that the integrals of 

sn 2 u, cn 2 u, dn 2 u, us 2 u, ds 2 u, cs 2 i&, . . . 

require for their expression new functions called elliptic in 
tegrals of the second kind, such as occur for instance in the 
rectification of the ellipse. 

For if, in the ellipse (cc/a) 2 +(2//6) 2 = l, 
we put x = a sin 0, y = b cos < ; 



then - = + = a 2 cos 2 </> + &2sinV = d\l - e 2 sin 2 0) ; 
Cup ct(p cicp" 

so that -=/J(l-e 2 sin 2 <j>)d<p =/A(</>, e)^-/dn 2 itcZ 

a o oo 

on putting = am(u, e); and 6, the excentricity of the ellipse, 
is now the modulus. 

The integral y^/(l /c -2 sin 2 0)(i0 oryX(0, K)d<f> is denoted by 

o 

JE((p, K) by Legendre, and called the elliptic integral of the 
second kind ; and when the upper limit is JTT, the integral is 
denoted by E I K, or by E simply, and called the complete elliptic 
integral of the second kind. 

Examples. The following examples are collected chiefly 
from Legendre s Functions Elliptiques ; the results, being 
now expressed by the inverse elliptic functions, will serve as a 
guide to the substitutions required to reduce the integrals to 
the standard elliptic forms, and the correctness can be tested 
by differentiation as an exercise. 



4. /(x-a.x-/3)- 



idx 



THE ELLIPTIC FUNCTIONS. 



6. 



7. Prove that, if w n = 4>x n (I - x n ), 
2- 1 / 



and express the result when n = 3, 4, or 6. 

8. Prove that, if x a is a factor of the cubic X, so that 

X = (x - a )(ax 2 + 2bx + c) ; 

fv 7. 3 

7^ ~aa 2 

a 

an integral occurring in the determination of the motion of a 
projectile in a resisting medium. 

Evaluate the integral when acr + 2ba + c = 0, so that 



. Prove that (i.) A cn ^ = /1 
k y l + dnu A/l 



-dnu 



x.. . r K snudu _ 1 

J dnu + K K (!+K ) 

o 

/*8JC 

(iii.) /u sn%<iu = 2K(K-E)/ K 2 . 

o 

/-Jir 1 

(iv.) /f\<A, /c)sin 6<i^> = - sin -^ 

J K 







10. Prove that 

^//c /2 >K>E> 

11. Denoting the integral f(^)- n d^ by u n , establish the 

formula of reduction 

/c % w+2 - (n - 1) (1 + K* 2 )u n + (n- 2)u n _ 2 = - /rsin cos 
Evaluate -u w for -w. = 2, 3, 4, . . . . 



G.E.F. 



CHAPTER III. 

GEOMETRICAL AND MECHANICAL ILLUSTRATIONS 
OF THE ELLIPTIC FUNCTIONS. 

78. Graphs of the Elliptic Functions. 

Now that the Elliptic Functions have been defined and a 
few of their fundamental properties have been established in 
Chapter I. in connexion with the pendulum; while in Chap 
ter II. the reductions of the elliptic integral to the standard 
form have been tabulated, let us consider some further applica 
tions, and first in connexion with the graphs of am u, en u, 
sn it, dn u, represented by curves whose equations are of the 
form y am x, en x, sn x, or dn x. 

The graphs of these equations are given in fig. 5, in curves 
(i.), (ii.), (iii.), (iv.) ; the modular angle employed is 45, so that 
the curves can be plotted from the numerical values given in 
Table II., analogous to the graphs of the circular and hyper 
bolic functions, given in Chrystal s Algebra, Part II. ; thus, 
for instance, the curve y = &mx is the graph of the relation 
between $ and u in 5. 

We notice from the equations of 57, Chap. *IL, that by 
sliding the curves along Ox through a distance K, the curve 
y = snx becomes changed into y = sn(I -f #) = cn#/dn# or cdx, 
and not into y = cnx ; while the curve y = cnx becomes changed 
into y = cn(x K) = KS n.x/dux or tc sdx, and not into y = snx; 
so that the curves y = sn x and y = cnx are essentially distinct 
curves, and cannot be superposed, like y = cosx and y = sinx. 

The curve (i.), the graph of am x } consists of a regular un 
dulation, running along the straight line y = \irx\K\ so that 
am x = ^7rX/K-}- periodic terms = \-wx\K- 



THE ELLIPTIC FUNCTIONS. 



67 



in a Fourier series, where the B s are to be determined sub 
sequently ; and then by differentiation, 

dn x = (k-jr/K) { 1 + 22m,B n cos(nirxlK) }. 

So also the graph of E$ or Eamu, the elliptic integral of 
the second kind ( 77) consists, like (i.) the graph of aino:, 
of an undulation running along the straight line y = Ex/K; 
so that we may write, in Jacobi s notation, 



where Zx is a periodic function of x, which can be expressed in 
a Fourier series 



and then, by differentiation, 

dn 2 # = EjK+(irlK)I,nC n cos n-n-x/K ; 
whence also the expression for sn 2 # and cu-x in a Fourier series. 




Fig. 5. 

We proceed now to some mechanical and geometrical appli 
cations of these curves. 

79. PROBLEM I. The curve assumed by a revolving chain 
We shall prove that 

y/b = sn Kxlci 

(fig. 5, iii.) is the equation of the curve of a uniform chain, 
rotating steadily with constant angular velocity n about an 
axis Ox, to which the chain is fixed at two points, 2ct feet 
apart, gravity being left out of account, e.g. a skipping rope. 



68 ILLUSTRATIONS OF 

Denote by t the tension in poundals of the chain at any 
point, and by w the weight in Ib. per foot of the chain. 
Then the equations to be satisfied are 

d / dx\ _ . d / dy 
ds\ ds/~ ds\ ds. 

Therefore tdx/ds = T,a, constant, the thrust in poundals in 
the axis due to the pull of the chain ; and therefore 

rJ /rlii\ /n^-fin fl^ti rJw 

(.(/ / (A/U \ Iv W tt/t/ U/JU 



the differential equation of the curve of the chain. 
But 14-^ 2 

l 



so that dydty dsd?8 

dx dx z dx 

and therefore 

Integrating, supposing y = b when dy/dx = and ds/dx 

ds n 2 iv., 9 9 . 

^=l + w (^-^); 

so that t = Tds/dx = T+ i% 2 w(6 2 - y-). 



so that x is an elliptic integral of y, of the form (5) in 
Chap. II. ; and y is an elliptic function of x, obtained by 
inverting the function of the integral. 

To obtain this function, let y = b sin < ; then 



ri 2 wb- 



,1 T rX i_ /c nw 

so that d> = am K-, where = ^ ; 

a a 2T 

and y/b = sn Kx/a, 

the equation of the curve formed by the chain ; and now 2a 
denotes the distance between the ends of the chain. 



THE ELLIPTIC FUNCTIONS. 59 



We may denote Tj\n 2 w by Ji 2 ; and now 



h- K " 2 _ti 2 lab , la 

. , 19. ~?~ro J^K 



whence the modulus K and quarter period K can be determined 
when h and a are given ; and 



while ^ = ~A0; 
fix a 

and integrating, with the notation of 5 and 77, 

Kh* 



If 2^ denotes the length of the chain, then s = l when = 1^, 
and J^(0, *)=. , j^(0, /c)=^; and therefore 

I + a = JJ?JST^/ = ^ = 2a Ay^ /2 , 

from which /c, -ST, and .fi 1 must be found by a tentative process, 
from Legendre s F.E., II., Table II., when a and I are given. 
For instance, if K = K = |^/- as ^ n ^ a ble II., page 11, 

A"= 1-85407, ^ 
and 6/a = 1 5255, l/a = l 92 

80. When the chain is fixed at two points not in the axis, 
nor in the same plane through the axis, the chain when re 
volving in relative equilibrium will form a tortuous curve, 
which will sweep out a surface of revolution, of which the 
preceding curve y/b = snKx/a is a particular case of the 
meridian curve, while the general equation is of the form 



For in this more general case the equations of relative 
equilibrium are now 

d (,dx\ d ( dy\ , d (,dz\ 

-y-U-f ) = 0, -^-i t-Y- } + n- W y = 0, -y-f t-j- + n*wz = 0. 

ds\ ds/ ds\ dsJ ds\ dsJ 

Three first integrals of these equations are 



H, a constant; (2) 

and t-\-^n-iv(y 2 + z 2 ) = \, a constant (3) 



70 ILLUSTRATIONS OF 

Putting y* + z z = r z , 

th dy dz _ .dr 2 

y dx ^dx~^dx 

and from (1) and (2), 2/^-^| = f ; 

therefore, squaring and adding, 
d < 



d& 

or 



= 2 (|! - 4) - -*^* = ^(2X - 



suppose ; arid for r 2 to lie between b 2 and c 2 , we must suppose 
<i 2 > fr 2 > r 2 > c 2 , and as it is of the form (17), p. 37, we put 



W - r 2 = (6 2 - c 2 )cos 2 0, r 2 - c 2 = (6 2 - c 2 )sin 2 0, 
cZ 2 - r 2 = d z - c 2 - (6 2 - c 2 )sin 2 = (d z - c 2 ) A 2 0, 
where /c 2 = (6 2 -c 2 )/(rZ 2 -c 2 ). 

Then 



- c 2 )cos 2 sin 2 



or . r = 

so that <p = am Kx/a, 

where K z /a z = n*w 2 (d 2 - c 2 )/4T 2 = 4(cZ 2 - c 2 )/h 4 ; 

and then r 2 = y 2 + z 2 = b^r^Kxja + c 2 cr\ 2 Kx/a, 

the equation of the surface swept out by the chain, the meridian 

curve being similar to curve (iv.) in fig. 5. 

81. The chain will obviously take up the form which, with 
given length between the two fixed ends, has the maximum 
moment of inertia about the axis of revolution ; and we have 
thus investigated the solution of an interesting problem in the 
Calculus of Variations. 



THE ELLIPTIC FUNCTIONS. 71 

The form of the chain for a minimum moment of inertia is 
obtained by supposing that r 2 >d 2 , as in (13), p. 35 ; and by 
putting r 2 d 2 = (d 2 6 2 )tan 2 ^>, 

T 2 -b 2 = (d 2 -b 2 )sec 2 <j>, 
r 2 - c 2 = (d 2 - c 2 ) A 2 sec 2 0, 

K 2 = (b 2 - c 2 )/(d 2 -c 2 ), as before. 

Then ~ 2 = 4(d 2 - 6 2 ) 2 tan 2 






- b 2 ) 2 (d 2 - c 2 )tan 2 sec 4 A 2 0, 



so that <p = am Kx/a, 

and then y 2 + z 2 = d 2 sQC 2 <j> - 6 2 tan 2 



a - b 2 sc 2 Kx/a 

is the equation of the surface of revolution upon which the 
chain lies, when its moment of inertia about the axis of x is 
a minimum. 

The projection of the chain upon a plane perpendicular to 
the axis is to be investigated subsequently. 

82. When the two points to which the ends of the chain are 
fastened lie in the axis, or in a plane through the axis, the 
chain takes the form of a plane curve, whose equation is 

y/b = sn Kx/a 

for a maximum moment of inertia, as already shown in 79 ; 
and y en Kxja = cZ, or y = d nc Kx/a 

for a minimum moment of inertia ; which can be proved as a 
simple exercise in the Calculus of Variations, by considering 
the variation of the integral 



83. PROBLEM II. " The curve on which an ellipse, of semi- 
axes a and b, must roll for its centre to describe a straight line 
Ox is the curve whose equation is 

y/a = dn x/b, 
the modulus /c being the excentricity of the ellipse." 



72 



ILLUSTRATIONS OF 



For if the centre M of the ellipse describes the horizontal 
straight line Ox (fig. 6), M must always lie vertically over P, 
the point of contact with the fixed curve, so that the ellipse 
rests in neutral equilibrium if its centre of gravity is at the 
centre M\ teeth being cut in the curves, if requisite, to prevent 
slipping. 

Therefore the polar subnormal 

^ n dr . ,. ... 1 cos 2 # sin 2 $ 

MG= " m the elhpse ~^~^~~ 



dO 
must be equal to the subnormal 

= y^r i* 1 the fixed curve AP } where MP = r = y. 




Fig. 6. 

Now in the ellipse, differentiating, 

2 dr (I 1\ . . I/I 1 1 1 

-- * ^Z == lf2 -- o)2sm0cos0=2 A /( 9 -- 5-r i 

r 3 dO \6 2 a 2 / \ \r 2 a 2 6 2 r 2 



snce 



or 



- 9 
r 2 a 2 



^ 
o 2 



7^ 2 
6 2 a 2 / 



so that in the fixed curve AP 



. dy 



by (9), p. 33 ; or, by inversion of the function, 

y/a = dn x/b. 



THE ELLIPTIC FUNCTIONS. 73 

The arc of the rolling carve is obviously the same function 
of r as the arc of the fixed curve is of y ; and therefore the 
arcs are expressible by elliptic integrals of the second kind. 

The curve AP can be described as a roulette, by a point P 
fixed to a certain curve which rolls on Ox, and therefore 
touches Ox at G, since G, the foot of the normal PG, is the 
centre of instantaneous rotation. 

Since PM is the perpendicular from a pole P on the tangent 
of the rolling curve, and that the relative orbit of P and M is 
the ellipse, therefore the pedal of the rolling curve with respect 
to the pole P is an ellipse ; or, in other words, the rolling 
curve is the first negative pedal of an ellipse with respect to 
its centre, that is, the envelope of lines drawn through each 
point on the ellipse perpendicular to the line joining the point 
to the centre of the ellipse. 

The first negative pedal of an ellipse with respect to its 
centre is called Talbot s curve ; its (p, CD) equation is 
1 _ cos 2 co sin 2 o> 
p~~~oT ~P 
and it is of the sixth degree (Cayley, Proc. R. S., 1857-9, p. 171). 

84, For a rolling hyperbola, changing the sign of b 2 , the 
fixed curve must be given by 

abdy ab 



r_ 

-J J (y *- 






by (8), p. 33 ; so that, by inversion of the function. 

a/y = en X/CLK, or y/a = nc X/CLK, 
is the equation of the fixed curve for the hyperbola. 

85. When the fixed curves are of the form of curves (ii.) and 
(iii.) in fig. 5, we shall find in a similar manner that the rolling- 
curves which will rest upon them in neutral equilibrium are 
given by 

1 cosh 2 , sinh 2 # 1 cosh 2 sinh 2 

_ - ; _ j _ /"1 1* __ - - _ _^ _ 

r 2 a 2 b 2 r 2 a 2 b 2 

Taking the first of these two rolling curves, 



74 ILLUSTRATIONS OF 

_ dr _ ? V(a 2 - r 2 . b 2 + r 8 ) 

d0~ ~^T~ 

so that in the corresponding fixed curve 



dx ab 

x -f a M y ab fy 

Jj(a*-y2.b 2 + y 2 ) T^hP) U 
by (7), p. 33 ; so that, by inversion, 



y/a = en #/&/c, with mod. K = a/^/(a 2 + 6 2 ). 
Similarly it can be proved that the second rolling curve can 
rest in neutral equilibrium on the fixed curve (fig. 5, iii.) 
y/a = sn xja, with mod. a/b. 

86. TROBLEM III. Dynamical Problem. "The curve 
rcuO = c is the relative orbit of the centres of gravity of a 
straight rod fitting into a smooth straight tube, resting on a 
smooth horizontal table, when struck by an impulsive couple, 
the centres of gravity of the rod and of the tube being initially 
c feet apart." 

Suppose the rod to weigh m Ib. and the tube to weigh 
M Ib., and denote the moments of inertia about the centres 
of gravity by mk 2 , MK 2 (Ib. ft. 2 ). 

Then, if P is the C.G. of the rod, Q of the tube (PQ = r), and 
the (stationary) C.G. of the system, 



Denoting by n the initial angular velocity communicated to 
the system by the impulsive couple, then from the Principle of 
the Conservation of Angular Momentum, 

{m(k 2 + OP 2 ) + M(K 2 + OQ*)}(de/dt), 

( 79 MY mMr 2 \dO ( 72 , ,, T , 9 , mMc 2 \ /1X 
or 1 ^+XK*+--= m ] l *+MK*+n ..... (l) 



Again, from the Principle of the Conservation of Energy, 



m 



or, after reduction, 
1 

2 



THE ELLIPTIC FUNCTIONS. 75 

the kinetic energy in foot-poundals, is constant, and 



Therefore, employing the value of dOjdt given by (1), 



or, finally, 



+ J/ZT 2 + mJ/c 2 / (m + Jl/) f 
so that r is an elliptic function of 0, given by (8), p. 33. 
We therefore put r = csec<; and then find 



where /c 2 = ^ T777- 

so that = am 0, cos < = en ; and therefore 

r en = c. 

87. When c = 0, /c = l, and this method fails; but now 

r 2 dfl 2 = (ra& 2 + MK *)(M+ m) = a 2 
suppose, where a 2 = (991 + M)(mk z + MK*)/mM ; 

/^ ft K* st 

and now = 7 77 rr-^ = sinh~ 1 -, 

J T^/( I -f r 2 /a 2 ) r 

or rsinh$ = , 

the equation of one of Cotes s spirals, the relative orbit of the 
centres of gravity of the rod and tube, ultimately described 
after leaving the unstable position of coincidence. 

The system of the rod and tube may be supposed started 
by any arbitrary impulse, not necessarily a couple, and the 
essential character of the relative motion is unaltered; but now 
the C.G. of the system is no longer at rest. 

88. Other mechanical arrangements, leading to the same 
equations of motion, will readily suggest themselves ; thus the 
tube may be supposed to be one of the hollow spokes of a 
wheel of weight M lb., moveable about a fixed vertical axis, 
while the rod is one of a number of equal rods, or balls, of 
collective weight m lb., one in each tube, and initially placed 
with the C.G. at a distance c from the axis of the wheel. 



76 ILLUSTRATIONS OF 

Now, if the wheel is started by an impulsive couple with 
angular velocity n, the path of the C.G. of each rod or ball in 
its spoke will be of the form 

r en (9 = c. 

89. PROBLEM IV. Central Orbits and Catenaries expressed 
by Elliptic Functions. 

When a Central Orbit, expressed in the polar coordinates 
(1/u, 0), is described under an attraction to the pole, of magni 
tude P (dynes per gramme), then, as is proved in treatises on 
Dynamics, P is given by the equation 

73 z.2 *fd Zu i \ i 7 40 1 dO 

P = h2 u 2l + u \ W h ere / 6==r 2 = 

\d0 2 J dt u 2 at 

and the constant h is twice the rate of area swept out by the 
radius vector ; and v the velocity is given by 

h 2 ,jd 



Given the equation of the orbit as a relation between u and 
0, the value of P as a function of u is thence easily determined 
by differentiation, as in 30 ; let us then determine P for the 
orbits au = sn, en, tn, or dn m6 ; 

also for the inverse curves 

au = ds, nc, cs, or nd m9, 
in Glaisher s notation ; the remaining orbits 

au = cd, sd, dc, ds mQ ; 

are not distinct curves, being merely formed by reflexion in the 
line 0=$K/m, since cd mO = 8i\(K mO) ( 57), etc. 

As in 30, we shall find by differentiation that (d 2 u/d6 2 ) + u is 
always of the form Au + Bu s , so that P is of the form /x,u 3 + vu 5 ; 
and conversely, given this form of P, we find by integration 
that (du/dO) 2 is of the form C+Du 2 + Eu* , so that is an 
elliptic integral of u, and u an elliptic function of (9, of which 
the results are given in 36. 

When the orbit is given by 



we find by differentiation, as in 30, that P is of the form 
Xu 2 + /xu 3 + j/u 4 ; and conversely, when P is of this form, 
(du/dO) 2 is a cubic form in u ; and 6 is given as an elliptic 
integral or inverse elliptic function of u, by the results of 
equations (12) to (45), Chap. II. 



THE ELLIPTIC FUNCTIONS. 77 

As an exercise the student may determine the value of P 
and v\ as functions of u or r, in the orbit 
1 _cn 2 m$ sn 2 ra0 

and its inverse curve, whose equation is of the form 

Similarly the central forces required to make a chain assume 
the form of one of the preceding curves can also be determined 
(Biermann, Problemata quaedam mechanica functionum 
ellipticarum ope soluta, Berolini, 1865). 

When a transverse force T is introduced into the field of 
force, then h is no longer constant, but, as demonstrated in 
treatises on Dynamics and the Lunar Theory, - 
dh?_2T T _dlogA. 



,., 

dP+ -~ 

*= 



V~ dO 
T du 



If we assume P = h 2 u 3 ; then 

d (, du\ dlosh ,du 

l s+ ~ => or h = c > a constant 



But = /m 2 , so that = Cn 2 } or =-C, which shows 

that the body approaches the centre with constant velocity C. 
Suppose, for instance, we take an orbit given by 

m6 = am au, 



then h = C = C ~ 
and P = hW = 



T= u*~ - = - CAin mO cos mO ; 
so that V, the potential of the field of force, is given by 



and then P= _, T= -. 

- 



78 ILLUSTRATIONS OF 

90. PEOBLEM V. The motion of Watt s Governor. 

"The oscillations of Watt s Governor between the inclina 
tions a and /3 to the vertical, when constrained to revolve with 
constant angular velocity co, are given by 

tan|$ = tanjadn(?i, K), with K = tan J/3/tan Ja, 
where 6 denotes the inclination of an arm to the vertical axis 
at the time t." 

Consider the motion of either rod and ball, as if unconstrained 
by the other, and denote by C the moment of inertia of the 
rod and ball about its axis of figure, and by A the moment of 
inertia about the axis on which the rod turns at the upper 
joint (fig. 7). 




Fig. 7. 

Drawing the three principal axes OA, OB, OC at 0, and 
three moving coordinate axes Ox, Oy, Oz, such that Ox 
and OA are coincident, Oz is vertical, and yOz, BOC in 
the same vertical plane, then the components of angular 
velocity about OA, OB, OC are (dO/dt), o>sin$, e*>cos$; 
and the corresponding components of angular momentum are 
-A(d6/dt), -4o>sin0, Coo cos 0. 

The components of angular momentum about Ox, Oy, Oz 
will therefore be 

fc 1= -A(d6/dt) t h 2 = (C-A)>sin6cos6, h s = (Ccos*6+Asin*6)a>; 
while the component angular velocities of the coordinate axes 
Ox, Oy, Oz are ^ = 0, 6. 2 = 0, 6 3 = u>, with the notation of 
Routh s Rigid Dynamics. 

Take the poundal as the unit of force, and denote by M the 
weight in Ib. of either arm and ball, by h the distance in feet 
from of the centre of gravity; the equation of motion 



THE ELLIPTIC FUNCTIONS. 79 

obtained by taking moments about Ox or OA is 



or -A(d 2 0/dP) + (A-C)u 2 sin0cos0 = Mghsin0 , (1) 

so that, if A = C, the motion reduces to simple pendulum motion. 
Integrating, on the supposition that a > > j8, and that 
dO/dt = Q when = a and /3, 

rZ6 2 A G 

-jp = -j or(cos cos a)(cos /3 cos 0) (2) 

The position of relative equilibrium is given by d 2 0/dt 2 = Q; 
and then, if 9 = y, 

so that in these oscillations the point D, which controls the 
valve, makes equal excursions above and below its position of 
relative equilibrium. 

The technical name for these oscillations is "Hunting"; and 
some kind of frictional constraint is required to prevent these 
oscillations from becoming established. 

(Maxwell, Proc. R. S. t 1868.) 

Denoting tanja, tan J/3, tanj$ by a, b, x respectively, then 
equation (2) may be written 

4 dx 2 _A-G pfl-x* l-a 2 \/l-6 2 1-; 

~dP~ ~A~ w \T+^~T+~ 



or = cos a cos 

and this, by equation (9), p. 33, gives 
x = adn(nt, K), or tanj$ = 
where /c = 6/a = tan J/3/tan Ja, and ?? =o)sinjacosj/3 x /(l G 
For a small oscillation, we put a = /3; and then /c =l, /c = 
and now the period of an oscillation 

27T 47T 



/_J L 

\ A 



^ to sn a 

91. If we suppose the whole weight of a rod and ball con 
centrated at the centre of gravity, we have (7=0, A=Mh-\ 
and now the motion may be assimilated to that of a particle 
in a smooth circular tube, which is made to rotate about a 
vertical diameter with constant angular velocity w. 

(Prof. B. Price, Analytical Mechanics, 403). 



80 ILLUSTRATIONS OF 

The equation of motion (1) now reduces to 
h .. ., AcoVm cos = g sin 0, 

where h denotes the radius of the circle ; and for oscillations 
on one side of the vertical between a and /3, a > 9 > ft, 

(dO/dt) 2 = o) 2 (cos - cos a) (cos ft - cos 0), 
the solution of which is, as before, 

tan J0 = tan Ja dn nt, 
where /c = tan J/3/tanJa, n = w sin | a cos J/3. 

If the particle in its oscillations just reaches the lowest 
point of the circle, ft = ; and then K = 0, /c = 1 ; and now 
dnnt degenerates into sechnt ( 16) ; so that 

tan J$ = tan \a sech nt, where n = o> sin Ja ; 
the position of relative equilibrium being given by 

cos y = g/w 2 h = |(1 4- cos a) = cos 2 Ja. 

If the particle passes through the lowest point, it will come 
to rest again where 0= a; and now 

(dO/dt) 2 <o 2 (cos cos a)(2 cos y cos a cos 0), 
where 2 cos y cos a > 1; and the solution of this equation is 

tan J0 = tan Ja en nt, where n = ^^/(cos y cos a). 
When a = 7r, we shall find the motion given by 

so that, after an infinite time, the particle just reaches the highest 
point of the circle, where it will be in unstable equilibrium. 

A still greater velocity of the particle relative to the tube 
will make the particle perform complete revolutions, which 
will be expressed by 

ta,uW = Ctnnt. 

We have supposed the circular tube to be made to rotate 
with constant angular velocity about a vertical diameter ; but 
the motion of the particle relatively to the tube will be found 
to depend on similar equations when the tube is attached in 
any other manner to the vertical axis. 



THE ELLIPTIC FUNCTIONS. 81 

92. Such will be the motion of a pendulum swinging about 
an axis fixed to the Earth, and now it is interesting to notice 
other cases of motion of bodies which can be directly compared 
and made to synchronize with the motion of an ordinary 
pendulum, swinging through a finite angle. 

Thus the pendulum, if moveable about a smooth vertical 
axis, which is fixed to a wheel moveable about a fixed 
vertical axis, the inertia of the wheel being sufficiently great 
for the reaction of the pendulum to have no sensible effect on 
its angular velocity, will perform pendulum oscillations, with 
g replaced by aw 2 , o> being the angular velocity of the wheel 
and a the distance between the axis of the wheel and of the 
pendulum. 

Again a cylinder of radius a and radius of gyration k, rolling 
inside a fixed horizontal cylinder of radius b, will synchronize 
with a pendulum of length l = (b a)(l+& 2 /a 2 ). 

If the fixed horizontal cylinder is free to rotate about its 
axis, and has its centre of gravity in the axis, then the length 
of the equivalent pendulum is 



-, , 
l=(b-a)(I+n\ where n = - 2 l+- 2 



mk-, MK 2 denoting the moments of inertia about the axes 
of the rolling and fixed cylinders. 

The rolling cylinder may be replaced by a waggon on 
wheels, and the motion can still be compared with that of 
a pendulum. 

A circular cone, whose C.G. is in its axis of figure, and whose 
axis is a principal axis, performs pendulum oscillations when 
it rolls on an inclined plane, or inside or outside another fixed 
cone, whose axis is sloping, the vertices of the cones being 
coincident; the determination of I, the length of the equivalent 
pendulum, in these cases is left as an exercise to the student. 

In those cases where the finite oscillations are not of the 
pendulum character, we suppose the motion indefinitely small ; 
and now, in small oscillations under gravity, instead of giving 
the formula for the period of a small oscillation, it is in general 
simpler to give I, the length of the pendulum, whose small 
oscillations have the same period. 



G.E.F. 



82 ILLUSTRATIONS OF 

Thus for the vertical oscillation of a carriage on springs, 
I is equal to the permanent average vertical deflection of the 
springs, due to the weight of the body of the carriage. 

For the small vertical oscillations of a ship, l=V/A, where 
V denotes the displacement of the ship (in cubic feet), and A 
the water line area (in square feet) ; and if the ship is floating 
in a dock of area B sq. feet, then it is easily proved that 



93. The Reaction of the Axis of Suspension of a Pendulum. 

It is important to know the magnitude of this reaction in 

the case of a large swinging body, like a bell in a church tower. 

Denote by X and Y the horizontal and vertical components 

of this reaction, considered as acting on the swinging body ; 

and take the gravitation unit of force, the force of a pound. 

Then X, Fand W t applied at the centre of gravity G (fig. 1), 
will be the dynamical equivalents of the motion of the body, 
collected as a particle at G ; and since the component accelera 
tions of G are h(dO/dt) z in the direction GO, 

and h(d 2 0/dt 2 ) perpendicular to GO, 
therefore, resolving horizontally and vertically, 

Wh(d 2 0/dt 2 )cos 9- Wh(dO/dt) 2 sin = Xg, 
Wh(d 2 0/dt 2 )siu 0+ Wh(dO/dt) z cos Q=Yg-Wg- 
while, from the pendulum motion, 

I(d 2 0/dt*) =-g sin 0, l\d0/dt) 2 =g(2R-lvQrs 6). 
From these equations we find 

Y ^ 4>Rh 2h 

pp = 1 jsa&O + ,2- cos -/-cos 0(1 cos Q) t 

Y , h (2h 

r ~W ~~ + I = ~ VI 

x i-2h 4Rh\ .. an . 

TJ/- = I ~i --- /2/ sln r sm cos 0, 
and therefore the resultant of X and F TF(1 h/l) is a force 



in the direction GO ; and T varies as the depth of P below 
the line 2/ = i^ + i^> 

whence X and F are easily constructed. 



THE ELLIPTIC FUNCTIONS. 



94. In the simple pendulum, h = I, and the tension T of the 
thread PO is given by 



At the end of a swing y = 2R, and T/W=l-2R/l; so that, 
if 2R is less than I, T is always positive. 

But if 2R is greater than I, so that the plummet swings 
through more than 180, T changes sign, and the thread will 
become slack, unless replaced by a light stiff rod. 

When 2R is greater than 21, the pendulum makes complete 
revolutions ; and now, at the top of a revolution, y = 21, and 
T/W=4<R/l 5 ; and when 2R is greater than -JZ, T is again 
always positive, and the plummet can be whirled round at 
the end of a thread, without the thread becoming slack. 

95. When the axis of suspension of the pendulum is hori 
zontal, and cut into a smooth screw of pitch p, the equation of 
energy gives 

W(V + k*+p*)(dOldt) 2 = Wg(H-h vers 0), 
if the centre of gravity descends from a height H above its 
lowest position ; so that 

( It,- + tf+p 2 )(d*Oldt 2 )=-ghsm 0, 
and therefore I = h + (k 2 + p 2 )/h ; 

and now in addition to X and F, the reaction of the axis exerts 

a horizontal longitudinal component Z and a couple pZ, given by 

W d 2 6_.-Wphsin6 

~ g P dt 2 ~ 2 



Similarly the increase in I due to the pendulum being sup 
ported on friction wheels may be investigated. 

As an exercise the student may investigate the small oscil 
lations of a system of clockwork, in which the wheels are 
unbalanced about the axes, and prove that for small oscilla 
tions the length of the simple equivalent pendulum is given by 

I = (2wfcy)/(Zti&p*eoB a), 

where w denotes the weight, wh the moment, and iuk 2 the 
moment of inertia of a wheel about its axis ; a denoting the 
angle which the plane through the axis and centre of gravity 
makes with the vertical in the position of equilibrium ; and 
p denoting the velocity ratio of the wheel. 



84 



ILLUSTRATIONS OF 



96. The Internal Stresses of a Swinging Body. 

These internal stresses are most forcibly realized on board a 
ship rolling in the sea, not only in their effects as producing 
sea-sickness, but also in causing the cargo to shift, if the cargo 
is grain, coal, or petroleum, in bulk. 

It is usual to consider the ship as acted upon by two forces, 
(i.) W tons, the weight or displacement of the ship, acting 
vertically downwards through the centre of gravity 6r, 

(ii.) W tons, the buoyancy of the water, acting vertically 
upwards through M the metacentre (fig. 8). 

t 




Fig. 8. 



These two forces form a couple of moment W.QM.smO 
(foot tons), so that the ship will roll about a horizontal longi 
tudinal axis through G, like a pendulum of length GL = k-jGM 
feet, Wk 2 denoting the moment of inertia of the ship about 
this axis of rotation. 

Now to find the force which acts upon w, any infinitesimal 
part at P of the ship, to give it its acceleration and to balance 
its weight, we refer the point P to axes Gx and Gy, drawn 
upwards through GM and perpendicular to GM. 



THE ELLIPTIC FUNCTIONS. 85 

This force will balance the reversed effective force of w at P 
and the effect of gravity on w ; and therefore, in gravitation 
measure, will have components 

w d*9 w fd&\* , 

~~ V "7/2 x \J+ ) ~^~ w cos ft parallel to Gx, 

w d 2 6 w fdO\ 2 

~"g X ~dP~ g V (di) +w sm P arallel to G V- 
If w is suspended as a plummet by a very short thread, the 
thread will take the direction of this force, and will therefore 
make an angle with Gx 

_ i9 s i n Q _ x(d z O/dt*) - y(dO/dtf 
gcos6 + y(d*0!dP) - x(d6/dty 2 

Supposing the ship to roll like a pendulum of length , 
through an angle 2a, then 

I(d 2 0/dt 2 )=- -g sin 0, and ^I(d0ldt) 2 =g(cos6-cos a) ; 
and by 8, 

dW/dP = - n 2 sm 6=- 2- /i 2 sin J0 cos J0 = - *n* K sn nt dn nt, 

(dO/df = 2n\cos - cos a) = 4?i 2 (sin 2 Ja - sin 2 J#) = 47iVcn%l 

At any instant the lines of reversed resultant acceleration 

will be equiangular spirals, of radial angle 0, round the centre 

of acceleration G as pole, the resultant acceleration at P being 

/* . r r 

g,- sin cosec 0, and the resultant effective force w-j sin 6 cosec 0, 

when we put GP = r, and I(d6/dty=g sinOcot 0; so that 
tan = (sn nt dn nt)/(2ic cn 2 n). 

Superposing the effect of gravity, the resultant lines of force 
or internal stress will be equiangular spirals of the same radial 
angle 0, round a pole J, the position of which is obtained as 
follows (fig. 8) : Draw LK perpendicular to GL to meet the 
horizontal line GK in K; describe the circle on GK as diameter, 
and draw KJ making an angle GKJ=<}> with GK; this will 
meet the circle in /. 

For the resultant effective force of w at P, being 

/= iv-rsm 6 cosec = WTTJ> 

making an angle with GP t will, when compounded with w 
upwards, and taking the triangle PGJ turned through an 
angle as the triangle of forces, have a resultant 

t = w.PJjGJ, making an angle with JP. 



86 ILLUSTRATIONS OF 

This will be the tension and in the direction of a short thread, 
from which w is suspended as a plummet at any point P ; and 
the deflection of this plumb line from its original mean direc 
tion in the ship will be a measure of the tendency of a body 
to slide or of a grain cargo to shift ; and to a certain extent of 
the tendency to sea-sickness at this point of the ship and at 
this instant of its motion. 

The tendency will clearly have its maximum value at the 
end of a roll, when dO/dt = 0, and = JTT, and then / coincides 
with K. (Prof. P. Jenkins, On the Shifting of Cargoes, Trans 
actions of the Institute of Naval Architects, 1887.) 

The plumb line at P will now set itself at right angles to 
KP, while the surface of water in a tumbler at P will pass 
through K ; and a granular substance at P will begin to slip 
if KP makes with its surface an angle greater than the angle 
of repose of this grain. 

Thus up the mast, at a distance a feet from G, water would 
be spilt out of a tumbler, or sand in a box would shift, by the 
rolling of the ship through an angle 2, which would not spill 
or shift, if the ship heeled over steadily, until an inclination /3 
(the angle of repose of the sand) was reached, given by 

tan /3 = (1 + a/7)tan . 

At the centre of oscillation L, where a=l, there is no 
tendency for the water to spill, and this shows that the motion 
of the ship is felt least by going down below as far as possible 
in the middle of the ship. 

In a swing the body is very near the centre of oscillation, 
so that ordinary swinging is very little preparation for the 
motion of a vessel. 

A swing to act properly as a preparation for a sea voyage 
should be constructed as in fig. 5, to imitate, in full size, the 
cross section of the ship, suspended at M ; arid now the varying 
effect of the motion can be experienced by taking up different 
positions on the deck, up the mast, and in the cabins, con 
structed in this swing. 

Sir W. Thomson proposes to find the axis of rotation of a 
ship and the angle through which the ship rolls by noting the 
direction of the plumb lines of two such plummets, suspended 



THE ELLIPTIC FUNCTIONS. 



87 



at two given points across the ship ; planes through the plum 
mets perpendicular to the plumb lines at the extreme end of a 
roll would intersect in K\ the horizontal plane through K would 
meet the median longitudinal plane of the ship in the axis G ; 
while the plane through K perpendicular to the median plane 
would meet it in L, whence GL, the length of the equivalent 
pendulum, and therefore the period of small oscillations could 
be inferred, as a check on this construction. 

Example. A rod AB, whose density varies in any manner, 
is swung in a vertical plane about a horizontal axis through A. 
Prove that the bending moment of the rod is a maximum at a 
point P, determined by the condition that the C.G. of the part 
PB is the centre of oscillation of the pendulum. 

97. PROBLEM VI The Elastica or Lintearia. 

The Elastica is the name given to the curve assumed by a 
uniform elastic beam, wire, or spring, originally straight, when 
bent into a plane curve (fig. 9) by a stress composed of two 
equal opposite forces T, on the assumption that at a point P 
at a distance y from the line of the applied stress the bending 
moment Ty is equilibrated by a moment of resistance B/p, 
proportional to the curvature l//o ; and the constant B is called 
the flexural rigidity of the spring (Thomson and Tait, Natural 
Philosophy, 611). 



O M G x 




B 



Fig. 9. 



Then Ty = B/p, or y p = B/T= c 2 , suppose ; 

and by KirchhofFs Kinetic Analogue, the normal of the Elas 
tica performs pendulum oscillations on each side of a perpen 
dicular to the line of stress, as the point on the curve moves 
with a constant velocity. 



88 ILLUSTRATIONS OF 

For, when the normal has turned through an angle 0, the 

curvature - = -=- = 9- 

p ds c 2 

and by differentiation 

d*6 1 dy 1 . 

~r~2 = ~9 7 ~? sm $> 
as 2 c 2 <is c 2 

which agrees with the equation of pendulum motion 

d 2 0/dP = - n 2 siu ft if 8/c = nt. 

Corresponding with the oscillating pendulum we have the 
undulating Elastica, intersecting the line of stress at an angle 
a ; and thus, writing s/c for nt in 8, 

sin JO = K sn s/c, cos JO = dn s/c, 
sin = dy/ds = 2/c sn s/c dn s/c, 
so that y = 2c K en s/c, 

measuring s from the point A, at a maximum distance from the 
line of thrust ; and a graduated bow might thus be employed 
for giving mechanically the numerical values of the en function. 
In the nodal Elastica corresponding with the revolving 
pendulum, 

= 2 am S/CK, sin = 2 sn S/CK en S/CK = dy/ds ; 
SO that y = 2(c//c) dn S/CK. 

In the separating case, K = 1, and y = 2c sech s/c ; and 

JO = amh s/c, sin JO = tanh s/c, tan JO = sinh s/c, etc. 
In the undulating Elastica 

/y /Y 

~ = cos = ^/(l - 4/c 2 sn 2 s/c dn%/c) = 1 - 2/c 2 sn 2 s/c ; 
and in the nodal Elastica 



= cos = ^(l - 4 sn 2 s/c cn 2 s/c) =1-2 sn 2 s/c ; 

so that x is given in terms of s by means of elliptic integrals 
of the second kind ( 77). 

A great simplification is introduced when K = K=^ M J 2- J the 
Elastica now cuts the line of thrust at right angles, and 

cos = cn 2 s/c = J2/ 2 /c 2 , 

which shows that this Elastica is the roulette of the centre of 
a rectangular hyperbola, rolling on the line of thrust. 

It is easily proved that in this curve the radius of curvature 
p is half the normal PG ; also that a chain can hang in this 
curve as a catenary, provided the linear density is proportional 
to (ncs/c) 3 ; this is left as an exercise for the student. 



THE ELLIPTIC FUNCTIONS. 89 

Wheri /c = 0, the undulating Elastica corresponds with small 
oscillations of the pendulum, and the Elastica is ultimately 
coincident with the line of thrust, the ordinate y varying 
as sins/c or sinx/c; and then the length of the beam, 
TTC = Tr^/(B/T), is the extreme length at which the straight 
form of the beam begins to become unstable under the 
thrust T. 

The nodal Elastica becomes practically a circle when /c = 0, 
corresponding in KirchhofFs Kinetic Analogue to the practi 
cally uniform revolutions of a pendulum when the velocity is 
indefinitely increased. 

The Elastica is also called Bernoulli s Lintearia, being the 
cross section of a horizontal flexible watertight cylinder, when 
filled with water, the free surface of which lies in the line of 
thrust Ox] for if t denotes the constant circumferential tension. 

t/p=wy, the pressure of the water, 
or yp = t/w = c 2 . 

It is also the profile of the surface of water drawn up by 
Capillary Attraction between two parallel plates (Maxwell, 
Encyclopaedia Britannica, Capillary Action). 

The student may prove, as an exercise, as in 80, that if the 
wire is bent into a tortuous curve by balancing forces and 
couples at its ends, it will assume the form of a curve in a 
surface of revolution defined by an equation of the form 



(Proc. London Math. Society, vol. XVIII.) 

98. PROBLEM VII. Sumner Lines on Mercators Chart. 

Sumner Lines, so called after Captain Sumner, of Boston, 
Massachusetts, are the projections on M creator s chart of 
small circles on a sphere ; if simultaneous observations are 
taken of the chronometer and of the altitude of the sun or a 
star, the observer knows that he must lie on a small circle 
having its pole where the Sun or star at that instant was in 
the zenith, and having an angular radius the complement of 
the observed altitude; and two such observations are em 
ployed in Sumner s Method for determining the ship s place. 

According as the observed altitude of the Sun or the star is 
greater or less than the declination, the small circle on the 



90 ILLUSTRATIONS OF 

Earth does not or does enclose the polar axis; and the cor 
responding Suraner line will be a closed or open curve, whose 
equation may be thrown into the form 

cosh y/c = sec a cos x/c, ....................... (i.) 

or sinh y/c = tan /3 cos x/c ....................... (ii.) 

On Mercator s chart ( 16) the latitude and the longitude 
of a point whose coordinates are x, y may be written 



where Trc/180 is the length on the chart of a degree of longitude 
at the equator. 

These relations are obtained by noticing that the bearing by 
compass of two adjacent points on the chart will be the same 
as on the terrestrial sphere, if 

dy_ dO 
dx cos 6d(p 

and now, if x = C(f>, so as to make the meridians of longitude 
equidistant parallel straight lines, then 

dy/dO = c sec 6, y/c =/sec OdO, 
or ( 16) $ = amh y/c. 

Now let <5 denote the declination of the Sun or star, y the 
observed altitude, < the difference of longitude of the observer 
and of the object ; then in the spherical triangle SPZ 



S denoting the Sun or star, Z the zenith of the observer, and 
P the pole of the Earth s axis. 

Since cos 8Z= cos PS cos PZ+ sin PS sin P^cos SPZ, 
therefore sin a = sin S sin 6 + cos S cos cos 0, 
or cos S cos (j> = sin a sec 6 sin S tan 

= sin a cosh y/c sin S sinh y/c ; 

and according as a is greater or less than S, this is reducible to 
the form A cosh(y b)/c or Bsiiih(y 6)/c; and this again 
by a change of axes to the form of (i.) or (ii.). 

(Crelle, XL, Gudermann, on the Loxodrome ; Messenger of 
Mathematics, XVI. and XX., Sumner Lines.) 
Differentiating equation (i.) with respect to x, 
dy _ sec a sin x/c _ sec asm x/c 
dx ~ sinh y/c /v /(sec 2 a cos 2 x/c 1) 
ds tan a _ sin a 

dx x /(sec 2 a cosPx/c 1) x /(sin 2 a sin 2 x/c) 



THE ELLIPTIC FUNCTIONS. 



91 



so that, as in 3, 4, and 8, 

sin x/c = K sn s/c, cos x/c = dn. s/c, 

cosh y/c = sin a dn s/c, sinh y/c = tan a en s/c, 
the modular angle being a. 

This shows that s/c in the closed Sumner Line (i.) may be 
equated to nt in the oscillating pendulum, and then x/c will be 
half the angle made by the pendulum with the vertical ; also 

in the Sumner Line 

dx 
cos \[s = -=- = cn. s/c, or i/r = am s/c, 

the intrinsic equation ; and p = c sin a sec x/c. 

The differentiation of equation (ii.) gives in a similar manner 



_ 

dx 

so that x/c = am s/c, with mod. angle /3 ; 

and now, in the corresponding undulating Sumner Line, x/c is 
half the angle made with the vertical by a revolving pendulum, 
if we put s/c = Knt. 



Also 



-T- 



, l//c) 



by 29 ; so that ^ = am(/cs/c, l//c), 

the intrinsic equation ; and p = c cosec /3 sec x/c. 




Fig. 10. 

The second curve, by a shift of origin a distance JTTC to the 
right, becomes sinh y/c = tan j3 sin x/c, 
and then it cuts at right angles the first curve (fig. 10) 
cosh y/c = sec a cos x/c. 



92 ILLUSTRATIONS OF 

For, differentiating these equations logarithmically, 

,->ydy ,x 

coth^ -/- = cot-. 

c dx c 

, y dy x 

tanh -- -^- = tan - , 

c dx c 

and therefore the product of the ~ s is 1. 

In fact .putting sec a coth a, the curves are derivable as 
conjugate functions from the equation 

x + iy = c amh(a / + i/3). 

99. PROBLEM VIII. Catenaries. 

" The catenary for a line density proportional to cosh s/a, 
where s is the length of the arc measured from the lowest 
point, is of the form 

tanh y/b = dn x/a, or dn x/b, 

according as a, the ratio of the tension in pounds to the density 
in Ib. per foot at the lowest point of the catenary is greater 
or less than b ; the Catenary of Uniform Strength being the 
curve in the separating case of a = b" 

The equation of the Catenary of Uniform Strength, in 
which the linear density or cross section is so arranged as to 
be proportional to the tension, is well known (Thomson and 
Tait, Natural Philosophy, 583) being 

evl b cos x/b = l, or e^/ & = sec x/b ; 
or as it may be written 

tanh \y\~b = tsn\^x/b. 

For if O-Q denotes the density in Ib. per foot, and cr 6 the 
tension in pounds at the lowest point A, cr the density and 
0-6 the tension at any other point P, at a distance s from A, 
measured along the curve, the equations of equilibrium of 
APare 

o-b cos \js = CT O &, o-b sin -^ =Ja-ds. 

Thence a- = o- sec \}s, andycrc/s = a- b tan i/r ; 

so that <r = <r 6 SQC 2 \lsd\js/ds = <r sec \]s, 

or ds/d-fi = b sec i/r, 

s =f b sec \fsd\fs = b cosh ~ ^ec \{s = b cosh ~ 1 o-/o- , 
o 

a- = o- cosh s/b. 



THE ELLIPTIC FUNCTIONS. 93 

We might therefore take a piece of uniform flexible and 
inextensible material, cut out from a plane piece by two 
catenaries, or modified catenaries, say y/c= cosh x/b, and 
hang it up in a catenary of equal strength. 

Also x =/cos i/rcZs ==/*bd\lr = 61^, 

y =Jsiu \fsds =jb tan ijsdty = b log sec i/r ; 
so that y/b = log sec x/b, or e y/b = sec x/b t 

the equation of the Catenary of Uniform Strength. 

But now suppose two supports at the same level to be made 
to approach or recede from each other ; the piece of cloth or 
the chain will hang in a different catenary. 

Denoting by <r ct the tension in pounds at the lowest point 
A, and by t the tension at P, then 

t cos \fs = ar Q a, ts m\fs =f&ds = or b sinh s/b ; 

dy b . , s 

so that p or -~- = tan \^ = - smh y , 

the intrinsic equation of the curve. 



or v- aMp 



an elliptic integral, of the form (10), p. 33; and putting p = tan 
d\[r _ //cos 2 \/r sin 2 \/A 

~- 



In the separating case, a = 6; and then x = b\fs, as in the 
Catenary of Uniform Strength ; the greatest possible span of 
a catenary of given material is therefore 7 rb = 7TT/ iv, where T 
denotes the tenacity of the material, in pounds per sq. foot, 
and w the density or heaviness, in Ib. per cubic foot. 

But with a > b, 

^ *)> where K = b/a; 



so that JTT -f \/r = am x/b, 

A dy en x/b 

and -- = i^u\h-= --- L 



C en x/b sn x/b 7 f K 2 sux/b snaj/6 7 
=/~ 2 n J -dx = /- I 

J sn 2 #/6 J l 



-- --- 

dx sn x/b 

x/b sn 

I 
dn 2 x/b 



or tan h y/b = d n x/b. 



ILLUSTRATIONS OF 
With a < b, 



so that \r = am x/a, 

-, cfo/ . sn x/a 

and -^=tanr= -- 4- 



j -- -, 

ewe en 05/a 



/sn #/a en oj/a 7 _ /*/c 2 sn cc/a en a/a 
- 1) - / -- Ct/X == / i - n - / - A> - 

cu z x/a J dtfx/a-K 2 



o 
a 



or tanh y/b = i --- r = dn(^T aj/a), 
dnx/a 

by 57 ; so that by a change of origin, taking the axis of y in 
a vertical asymptote of the curve, its equation may be written 

tanh y/b = dn x/a. 

(Compare Cayley, on A Torse depending on Elliptic Func 

tions, Q. J. M., XIV., p. 241.) 

100. In the catenary formed by an elastic rope or flexible 
wire, obeying Hooke s Law " ut tensio sic vis," we may still 
have p = sinh u ; but u is no longer proportional to the arc s. 

We use o- to denote the uniform density of the rope when 
unstretched, and s to denote the length of rope which stretches 
in AP to length s, a- Q b denotes as before the tension in pounds 
of the rope at the lowest point A, and CT O C is used to denote the 
modulus of elasticity of the rope in pounds ; so that, by 

Hooke s law, 1 = 1 -| -- . 

as Q o- c 

Then, as before, for the equilibrium of AP, 

t cos i/r = (r a, t sin i/r =fo-ds = or s Q) 



so that p = -^- -f = sinh u, 

dx o 

if we put s = a sinh u ; 

and then t = cr ox /(a 2 + s 2 ) = cr a cosh u. 

t \ds . a 2 



m , ds A $ \ds Q , a 58 , o 

Inen ^ =1J ^_y = acO shuH cosh-u, 

/-y/j/ \ -_ /7//Vo/ /* 

Ct/ Cv \ O^n^ / t-v tv O 



and 7 - = ^/( 1 + p 2 ) = cosh u, 

dx 



THE ELLIPTIC FUNCTIONS. 95 

,1 dx a 2 , 

so that -= = cH cosh u, 

du c 

-z- = a sinh u-\ cosh u sinh u. 
du c 

Integrating, putting a/c = h, 

s/a = sinh u + %h(u + cosh u sinh u), 
x/a = u + h sinh u, 
2//a = cosh u + J/i sinh%. 

For the corresponding points on the rope, when it is supposed 
inextensible, putting c = co , and h = 0, 

s /a = sinh u, xja = u, y /a = cosh u, 

giving an ordinary catenary ; so that the tangents are parallel 
at corresponding points of the catenaries of the elastic and of 
the inextensible rope. 

The terms depending on h, considered separately, define an 
ordinary parabola ; so that the catenary formed by an elastic 
rope is something intermediate to a parabola and a common 
catenary. 

101. PROBLEM IX. Geodesies. 

" Investigation of the geodesies on the Catenoid, the surface 
formed by the revolution of a catenary round its directrix, and 
on the Helicoid, into which it can be developed ; also of the 
geodesies on the Unduloid and Nodoid, the capillary surfaces 
of revolution, of which the meridian curves are the roulette 
of the focus of a conic section, an ellipse or hyperbola, rolling 
upon the axis of revolution." 

The simplest mode of determining a geodesic on a surface of 
revolution is to treat it as the path of a particle moving 
under no forces on the surface, considered as smooth, so that 
ds/dt is constant ; and then, since the reaction of the surface 
passes through the axis, r z dO/dt is constant ; and therefore 



= 6, a constant, 
ds 

r and denoting the polar coordinates of any point of the 

projection on a plane perpendicular to the axis Ox ; and thus 

ds- dx 2 . dr 2 . r 4 



96 ILLUSTRATIONS OF 

In the catenoid r/a = cosh x/a, 
so that 



dx a 

and therefore, in the geodesic, 

r 2_ a 2 dr 2 .dr 2 r* 



+ ~ 






dO* b 2 

We must distinguish the two cases according as b 2 ^ a 2 . 

When b 2 >a 2 , then r 2 >b 2 ; the geodesic osculates the circular 
cross section of radius b ; and we have 

r sn 6 fr, with K = a/b, 
as the polar equation of the projection of the geodesic. 

When b 2 < a 2 , then r 2 > a 2 ; the geodesic crosses the circular 
section of minimum radius a; and supposing it cuts the 
meridian here at an angle a,b = a sin a ; and now 
r sn($//c) = a, the modular angle being a. 

In the separating case, b = a and K = 1 ; and then sn(9 = tanh 6; 
so that r tanh = a 

is now the polar equation of the projection of the geodesic, a 
curve having r = a as an asymptotic circle. 

Generally in any geodesic on a surface of revolution, which 
cuts the meridian curve at a distance r from the axis at an 

angle v, sin y = r-- = ; 

A da r 

so that sin ^ varies inversely as r. 

102. Now suppose the catenoid is divided along a meridian 
curve AP, and again along the smallest circular section A A , 
and that this section AA is drawn out into a straight line, of 
length 2-7TC& ; the rest of the surface, if flexible and inextensible, 
will assume the form of a Helicoid, or uniform screw surface 
of pitch a, such that its equation is 

z = a$, 

taking the axis of z along the axis of the surface, and p, the 
polar coordinates of the projection of a point on a plane per 
pendicular to the axis ; and AP will become a generating line 
of the Helicoid ; this is proved geometrically, by noticing that 
the length of the helix PP f on the Helicoid is equal to the 
length of the circle PP f on the Catenoid. 



THE ELLIPTIC FUNCTIONS. 



97 



The surface being inextensible, and a circular cross section 
of the Catenoid becoming a helix on the Helicoid, it follows that 

r 2 d6* = p-d<f + dz 2 = ( P 2 + a 2 )d<j> 2 ; 
and since r 2 = p 2 + a 2 , therefore 6 = <. 




Fig. II. 

Therefore the equation of the projection of a geodesic on the 
helicoid is either of the forms 



or (p 2 + a 2 )sn 2 = b 2 = a 2 /* 2 , 

_ a dn 



The Catenoid is the surface of revolution formed by a 
capillary soap bubble film, when the pressure of the air is the 
same on both sides of the film. The surface is easily formed 
practically by dipping a circular wire into soapy water and 
raising it vertically ; and it is evident from mechanical con 
siderations that the surface is a minimum surface ( 31). 

The Helicoid, into which the Catenoid can be deformed, can 
be produced in the same manner by a film between two coaxial 
helical wires of the same pitch (C. V. Boys, Soap Bubbles). 



G.E.F. 



98 ILLUSTRATIONS OF 

These surfaces are particular cases of Scherk s minimum 
surface, whose equation is 

,y ,aj(x* + y*-b 2 ) ,J(x* + y 2 -b*) 

s^atan-^ + ataji-V^TK-T 2 . 2\ + b tanh " // *T - 2 , 2 > 
# &vO 2 + 2/ + ) ^/(a; 2 + y + a ) 

or 



reducing to the Catenoid when a = 0, and to the Helicoid 
when 6 = 0. 

The verification in the manner of 32 is left as an exercise 
for the student. 

103. The meridian curve of the Catenoid is the roulette AP 
of the focus of a parabola aG, the pressure of the air being the 
same on both sides of the film (fig. 12). 

But when the pressure of the air inside the film is increased 
or diminished, we find that the surface of revolution formed 
by the capillary film has as meridian curve BP or OP, the 
roulette of the focus of an ellipse or hyperbola, the first surface 
being called the Unduloid and the second the Nodoid. 

(Maxwell, Capillary Attraction, Encyclopaedia Britannica.) 

Denoting by y, y f the perpendiculars from the foci P, P on 
the axis Ox on which the conic rolls, then in the Unduloid 
BP, generated by the focus P of a rolling ellipse bQ, 

y + y = (PQ+ QP )COS ^ = 2a COS ^, 

and yy =W; 

so that 6 2 + y 2 = 2ay cos \js. 

If in the meridian curve BP of the Unduloid, we denote 
the radius of curvature by /o, and the normal PG by n, then, 
since b 2 + y 2 = 2ay cos ^ = 2ay*/n, 

1 6 2 1 

therefore - ^ 9 + s~ ; 

n 2a 2 2a 



and since cos \fs = -- + -, 

lay 2a 

differentiating, 

. d\!s ( b* 1 \dy 

sin y-f = ( - - - o - ) -/-, 

as \2a / ?/ 2 2a/as 



THE ELLIPTIC FUNCTIONS. 



99 




Fig. 12. 



or 



so that __ = _. 

n p a 

Then, if p denotes the excess over the atmospheric pressure 
of the air inside a capillary film, in the shape of an Unduloid, 
and t the tension of the film, 

p = t(-} = L: 

\n p/ a 

so that, if inside a Catenoid, the pressure is increased, the 
surface is changed into an Unduloid. 

If the pressure is slightly diminished by p, the surface be 
comes a portion of a Nodoid CP ; for now 



and in the meridian curve CP of the Nodoid, the roulette of 
the focus P of a hyperbola cR with foci P and P", 

y" y = (P"R RP)cos \fs = 2a cos \js, and yy = b 2 ; 
so that 6 2 y 2 = 2ay cos i/r = 2ay 2 /n : 

l = 6 2 1 

n~2ay 2 2a 



and 



100 ILLUSTRATIONS OF 

In the geodesic on the Unduloid, 

y 2 dO/ds = a sin y. 

supposing the geodesic cuts the meridian curve at an angle y 
at its maximum distance a from the axis; also a = a(l+6), and 
the minimum distance /3 = a(l e), so that a/5 = 6 2 , a-h/3 = 2a; 
and y lies between a and /3. 

Now, in the projection of the geodesic on a plane perpen 
dicular to Ox, writing r for y, so that tsm\[^ = dy/dx 
ds 2 dx 2 dr 2 dr 2 r 4 



or 



02 

a 2 sm 2 y 

and ? cos \/s = (b 2 -\- r 2 )/2a ; so that 



= L ffl+^m ^ 

1 4a 2 Aa 2 sin 2 



leading to integrals of the form (72) and (73), p. 52. 

We suppose first that /3 > a sin y, so that the geodesic crosses 
the minimum section of the surface, and therefore all the 
sections if produced ; and now with a > r > /3 > a sin y, we 
have, according to equation (72), 



~ 



J(a: 2 -r 2 .r 2 -/3 2 .r 2 -a 2 sm 2 7 ) 

1 _cn 2 m^_ L sn 2 m0 

r 2 a 2 /3 2 " 

Secondly, if a > r > a sin y > /3, then the geodesic osculates 
the circle of radius a sin y, and is limited by the convex part 
of the surface between two such circles ; and the equation of 
the projection of the geodesic is obtained from the above 
merely by interchanging a sin y and /3. 

In the separating case a sin y = /3 ; and then K = 1, m = tan Jy ; 
and the polar equation of the projection of the geodesic is 

1 = sech 2 mfl tanh 2 mfl 
r 2 rf~ ~p*~~ 

a curve having an asymptotic circle y = /3. 

The formulas are similar for the geodesies on the Nodoid. 



THE ELLIPTIC FUNCTIONS : 

104. Eulers Equations resumed. Poinsot s Geometrical 
Representation of the Motion of a Body under No Forces. 

We now resume these equations of motion, of which the 
solution by elliptic functions has been indicated in 32. 

By the Principle of the Conservation of Angular Momentum 
(Routh, Rigid Dynamics, Chap. IX.) the axis OC of the re 
sultant angular momentum G will be fixed in space ; and the 
direction cosines of this axis with respect to the principal 
axes of the body being 

Ap/G, Bq/G, Cr/G, 
the component angular velocity about OC will be 



Or 6r 

where, as before, T denotes twice the kinetic energy of the body. 

It is convenient to denote this component of angular velocity 
about OC by a single letter, say /m; and also to replace G and T 
by Z> M and D/x 2 , making T/G = ^ and G 2 /T= D; and then D will 
be a constant quantity, of the same dimensions as A, B, C. 

If / denotes the moment of inertia about the instantaneous 
axis of rotation OP, and if OP denotes the vector of the 
momental ellipsoid at 0, then /varies as OP~ 2 , so that we may 
put I=Dh 2 /OP 2 , where h is a new constant length. 

Now, if o> denotes the resultant angular velocity about OP, 

T-Jo) 2 , or D^ = DhVjOP\ 
so that the angular velocity CD varies as OP : and 



IUL CD p q r 

The direction cosines of the normal of the momental ellipsoid 
at P being proportional to Ax, By, Cz, or Ap, Bq, Cr, are 
therefore Ap/G, Bq/G, Cr/G ; so that OC, the axis of G, is 
perpendicular to the tangent plane at P ; and if OC meets this 
tangent plane in C, it follows that OC = h, so that the tangent 
plane at P is a fixed plane ; and during the motion the 
momental ellipsoid rolls on this fixed plane, called the in 
variable plane, with angular velocity proportional to OP. 

The curve traced out by the point of contact P on the 
momental ellipsoid is called the polhode, and the curve traced 
out by P on the invariable plane is called the herpolhodc ; 



ILLUSTRATIONS OF 

these names are due to Poinsot, as well as this geometrical 
representation of the motion. 

(Theorie nouvelle de la rotation des corps, Paris, 1852.) 
The equation of the momental ellipsoid may now be written 



while Ax/Dli, By/Dh, CzjDh are the direction cosines of the 
invariable line 00 ; so that 

AW + B*y* + C V = D% 2 . 

The polhode is therefore the curve of intersection of these 
two coaxial quadric surfaces, and therefore lies on the cone 



called the polhode cone ; and the projections of the polhode 
on the principal planes are therefore 

(A - B)By* + (A-C) Gz l = (A- D)Dh z , .... 

105. Denoting by v the component angular velocity of the 
body about the axis OH, where OH is equal and parallel to CP t 



and, by solution of these equations, 
A-B.A-C . 



or = i/ 2 + (l - -g) (I - Q) yu 2 = ^ - v,?, suppose ; 

B-C.B-A /- I>\- D 



C-A.C-B 2 / D\(. D 

-^^ 1 " 



and in these equations we may replace p, q, r, o>, /x, v by 
^, T/, 0, OP, /t, p, respectively, where p 2 = OP 2 h 2 . 
Example. Prove that 



and simplify 



THE ELLIPTIC FUNCTIONS. 103 

106. On the supposition that 

AT>BT> G 2 > CT, orA>B>D>C, 

r never vanishes, and the polhode encloses the principal axis (?; 
but p and q alternately vanish, so that i/ 2 oscillates in value 



If we put v - 2 = (~ - l){(l - ^)cos 2 6> + (l - 5 
/r \U \ & * o/ 



then 



A- 



Or* = 



We now find, on substituting in one of Euler s equations, 

-D . 



, d 2 n 9 (A-B)(D-C) . 
and ~- = - D^- . -- - sm 6 cos 6, 



the solution of which is of the form, as before in IS and 32, 

= a,m(nt, K ), 

,A-D.B-C A-B.D-C 

n ~ = D ^- and K - = 



the anharmonic ratio of A, B, D, C ; while 



giving ( 32) 



107. Quadrantal Oscillations. 

The oscillations given by a differential equation of the form 



are called quadrantal oscillations (Thomson and Tait, Natural 
Philosophy, 322), the system having two positions of stable 



104 ILLUSTRATIONS OF 

equilibrium given by = and = TT, and two unstable posir 
tions in the remaining quadrants, given by 6 = |-TT ; for 
instance, an elongated piece of soft iron in a uniform magnetic 
field, or an elliptic cylinder moveable about its axis in a cur 
rent of liquid performs quadrantal oscillations. (Q. J. M. t xvi.) 

When the system performs complete revolutions, the solu 
tion is ( 18) = &m(mt/K, K) ; 

but if it oscillates about the positions of stable equilibrium, 
given by 6 = 0, the solution is ( 29) 

0= am(m, l//c), 

or cos 6 = dn(mt/K, K), 

sin 9 = Kfm(mt/K, AC), 
where K is less than unity. 

The second solution will apply to the second state of motion 
in 32, where AT> G 2 > BT> CT, or A > D> B > G, and where 
p never vanishes, and the polhode encloses the principal axis A . 

108. Differentiating the equations of 105 with respect to t, 
du_ dv_A-B.A-G dp^B-G.B-A dg_G-A.G-B dr 
<a dt~ v dt~ BG P dt~ CA q dt~ AB ~ r di 
_B-C.C-A.A-B 

ABC ~ PqT 

or = - 4. . o, a 2 - a, 2 . W 2 - a> 2 . 



so that co 2 and t/ 2 are elliptic functions of t, of the form given 
by equation (15), p. 36. 

But, on reference to equation (A), p. 43, we see that 



if e a , e b , e c denote the roots of 4s 3 # 2 s # 3 = ; so that on 
comparison we may make 

proportional to $u e a , ^u e^ $)u e c ; 
or, symmetrically, we can put 



C T * = - m\A - B)($u - e c ) ; 
where the factor -m 2 is introduced for the sake of homogeneity, 



THE ELLIPTIC FUNCTIONS. 105 

m being of the dimensions of an angular velocity, such as p, q, 
r, CD, [*, v ; and now, on substitution in Euler s equations, 
du* B-C.C-A.A-B (B-C C-A A- 

~dt* = -ABC- - m =(^r+-jr+-c 

suppose; so that u = & constant nt. 

109. As in 32, we take A > B> C- and then 
(i.) when AT>BT>G 2 >CT, or A>B>D>G, 
r never vanishes, and we must take 

e c > e a > pu > e b ; 

so that ^ = e c , e 2 = e a> e 3 = e b ; 

(ii.) when AT> G 2 >BT> CT, or A >D>B>C, 
p never vanishes ; and then 



and we must take e 1 = e a , e, = e c , e. = e b . 

Since pu oscillates between e. 2 and e 3 , and is taken 
initially equal to e s , we find, on reference to equation (42), 
p. 45, that we must put 

u = 2^ + o> 3 nt, 
so that the constant of integration for u in 108 is 2^ + 0)3. 

Now, at the cost of symmetry, to get rid of the imaginary 
o> 3 , and to make the argument of the elliptic functions a real 
quantity nt, equation (42), expressed in the direct notation, 



gves 



B 

and e b always replaces e 3 , while e a replaces e v e c replaces e 2 , or 
vice versa, according as the polhode encloses A or (7. 

110. For the determination of e at e b , e c , we have the equations 

*a+ e b + e c = 0, 

(B-C)e a + (C-A)e b + (A - B)e c = T/m 2 = D^fm 2 , 
A(B- 0)e a + B(C-A)e b + C(A - B)e c = G*/m* = D^/m*, 
whence A T- G 2 = m 2 (C -A)(A- B)(e b - e c ), 

BT- G* = m *(A -B}(B- C)(e c - fl ), 



106 ILLUSTRATIONS OF 

A-D 



6 *~ 



B-D 



a m 2 A-B. B-V 
= D/x 2 C-D 

m*~B-C.C-A 

so that e c e a is taken positive or negative, according as 
BTG 2 or B D is positive or negative; while e b e c and 
e b e a are always negative, as explained above. 

Also (ea eb)-(< c-ta) = 3sa, > 

whence the values of e a , e b , e c . 

Then </ 2 = {(e 6 - e c ) 2 + (e e - e a )* + (e a - e fr ) 2 } 

can be found ; and the discriminant ( 53) 



= 

r_^2 3 _- 

~ ~~ 



111. We have supposed no forces to act; but the case in 
which the impressed couple is always parallel and proportional 
to the resultant angular momentum leads to equations which 
can be solved in a similar manner ; in this way we imitate the 
motion of a body, like the Earth, which is cooling and con 
tracting uniformly. 

Now, the component impressed couples about the principal 
axes being of the form \Ap, \Bq, XCr, 

A(dp/dt)-(B-C)qr = \Ap, ..., 
which, on putting p = e~ Xt p , and \t / = l e~ Xt , reduce to 



so that p , q f , r are the same functions of , which p, q, r would 
be of t, in the case where no forces act. 

In the case of the cooling and contracting body, we put 
A=e~ Xt A , B = e~ Xt B , C=e~ Xt C Q - ) and the equations become 



which are solved as before ; and Poinsot s geometrical repre 
sentation of the motion still holds, with slight modification. 



THE ELLIPTIC FUNCTIONS. 107 

A similar procedure will solve the following theorem : 
" A rigid body is moving under the action of a force whose 
direction and magnitude are constant, always passing through 
the centre of inertia (e.g. gravity), and of an absolutely con 
stant couple. 

" If p, q, T denote the component angular velocities about the 
principal axes at the centre of inertia, and if u, v, w denote the 
compound velocities of the centre of inertia along the principal 
axes at the time t ; then the determination of 

p/t, q/t, r/t, ujt, v!t, w/t, 

in terms of J 2 is the same as that of p, q, r, u, v, iv, in terms 
of t, when no forces act; t being reckoned from the commence 
ment of the motion." (W. Burnside, Math. Tripos, 1881.) 

112. To obtain the equation of the herpolhode, we notice 
that during the motion the polhode cone, fixed in the body, rolls 
on the herpolhode cone, fixed in space, being the common 
vertex ; corresponding areas of these cones are therefore equal, 
as also their projections on any fixed plane, for instance the 
invariable plane. 

Therefore if p, (p denote with respect to C the polar co 
ordinates of P on the herpolhode, 

dz d\ B dx dz Cz d dx 



Since *=0=?=fi 

p q r v 

therefore 



, 
dt h A 

dz d V ufA-B C-A , 



.1 



ABC 

D.C-D 



ABC 

which, combined with the value of di?/dt or dp 2 /dt of g 108, 



Pa" ~ P Pb~ - p~ 

will determine the equation of the herpolhode. 



108 ILLUSTRATIONS OF 

113. Using Weierstrass s functions of 108, 



M 2 1 

mJ 



M 

_ . C-A . J.- u 2 



xl a 

with Of = - 



^-(7 C-J. A-B 
- i i 



and then i w ""^ SSB1 - L j^- 1 (positive), 

/A 2 /l ^/^> !\ 

^-^ = l--l. (positive), 



and, since 6 X (or e e ) >$v> e 2 (or <? a ), 

we must, by (39), 54, where t is a proper fraction, take 

v = 

Therefore ^ 

at 



or 



and, integrating, 



and we are thus introduced to a new integral, called an 
elliptic integral of the third kind. 

The cone described in the body by OH ( 105) is called by 
Poinsot the rolling and sliding cone ; during the motion this 
cone rolls on an invariable plane through 0, while at the same 
time this plane turns with constant angular velocity JUL about 
OC ; so that, if p, $ denote with respect to the polar co 
ordinates of H on this plane, 



THE ELLIPTIC FUNCTIONS. 109 

114. With the notation of the elliptic functions of Jacobi, 
as in 106, 

p~ DD-C DD-C DA-D 



A-D.D-C D.A-B.D-C ^ 
-AC~ ABC 

which can be thrown into the form 



A 

DA-B 
on putting /c 2 sn 2 a = - -^ 

DB-C C B-D A B-D 

- a = BD=C> Cn " a= ~B D=C> dna = B A=D 

With e a = e z , e b = e 3 , e c = e lt and v = coj + t w s , then by (32), p. 44, 



and 

e b B AD 

so that a = K+t iK r . 

B-D 






_ 



B l-K 2 s 
i en a dn a n 



sna 
and, waiting u for nt, 

i sn a dn 



i en a dn a . /"/c sn a en a dn a sn 2 w 



sn a 

o 



. /"/c sn a en a 
u^/ y 5 

^y 1 /rsn-a 



7 
du, 



the last term an elliptic integral of the third kind, in the form 
employed by Jacobi. 

On putting sni6 = sin 0, and sn a = sin a, /c 2 sn 2 a= m, then 
,.cosaAa 



sn 

o 



the third elliptic integral, as employed by Legendre ; the 
further discussion of this integral must be reserved for a 
subsequent chapter. 



1 10 ILLUSTRATIONS OF 

EXAMPLES. 

1. Prove that, if the excentric anomaly in an undisturbed 
planetary orbit of excentricity e is represented by 2 am(u, e), 
the mean anomaly is 

2 am 



2. Prove that the envelope of the straight line rays 

K Z X sn u + (en u + K dn u)y = K sn u(dn u + /c en u) 
where u is the variable parameter, is the curve 



the caustic of parallel rays, after refraction at a circle, of 
refractive index l//c ; and find the order of this curve. 

(Cayley, Phil. Trans., 1857, " Caustics.") 

3. Prove that a portion of a flexible inextensible spherical 
surface of radius a, bounded by two meridians (a lune, or gore 
of a spherical balloon) can be bent into the surface of revolu 
tion given by 



, /c; 

0, $ denoting the latitude and longitude of the point on the 
sphere. 

Explain the geometrical theory, distinguishing the cases of 
K <1, and K >1. 

4. Denoting by o> the solid angle subtended by a circle of 
radius a at a point whose cylindrical coordinates are r, z with 
respect to the axis of the circle, prove that 

dco _ az K 3 
da~ 2arl K - 



Show how to determine the illumination at any point of the 
surface of the water at the bottom of a deep well, due to the 
light from the sky. 



THE ELLIPTIC FUNCTIONS. HI 

5. A uniform circular wire, charged with e coulombs, is 
presented symmetrically to a fixed insulated sphere of radius 
a centimetres, so that every point of the wire is at a distance 
/ cm from the centre of the sphere, the radius of the wire sub 
tending an angle a at the centre of the sphere. 

Prove that the electricity, in coulombs per cm 2 , induced at a 
point of the sphere whose angular distance from the axis of 
symmetry is 0, is given by 

E 



.,_ Wsinasinfl /2 _ a 2 - 2a/cos(fl - a) +/ 2 

~ a 2 - 2a/cos(0 + a) +/ 2 ~ a 2 - 2a/cos(0 + a) +f* 

6. Prove that if this sphere and wire gravitate to each other, 
and if the wire is free to turn about a fixed diameter perpen 
dicular to the line joining the centres, the wire will be in stable 
equilibrium when its plane passes through the centre of the 
sphere ; and prove that the oscillations of the wire due to the 
gravitation will synchronize with a pendulum of length 

) 

CJ cm 

where b denotes the radius of the wire, c the distance between 
the centres of the sphere and wire in cm, M the weight of the 
sphere in g, C the gravitation constant ; and 

JP= lj-X= J K (1 +O jr {(1 + K -)^K], 

where K~ = bc/(b + c) 2 . 

Determine the position of stable equilibrium and the length 
of the equivalent pendulum, when the attraction is changed to 
repulsion. 

7. Two uniform concentric circular wires of radii b and c cm, 
weighing M and M g, are freely moveable about a common fixed 
diameter. Prove that in consequence of their gravitation, the 
oscillations will synchronize with a pendulum of length 

Cm 
where F and K have the same values as before. 



CHAPTER IV. 

THE ADDITION THEOREM FOR ELLIPTIC 
FUNCTIONS. 

115. So far we have considered the elliptic functions of a 
single argument u ; but now we have to determine the for 
mulas which give the elliptic functions of the sum or difference, 
uv, of two arguments u and v, in terms of the elliptic functions 
of u and v ; and thence generally the formulas for the elliptic 
functions of the sum of any number of arguments u-\-v+w+...\ 
and the formulas for the duplication, triplication, etc., of the 
argument. 

The Addition Theorem for Circular and Hyperbolic 
Functions. 

The analogous formulas in Trigonometry for the Circular 
Functions are well known, namely, 

sin(u v) = sin u cos v cos u sin v, 
cos(t6 v) = cos u cos v + sin u sin v ; 
or, as they may be written, 

siu(uv ) = sin u sin i> sin% sin v, 
cos(u v) = cos u cos v + cos u cos v ; 
the accents denoting differentiation ; and to these*may be added 



. - 

1 + tan u tan v 

these formulas constituting the Addition Theorem for the 
Circular Functions. 

For the Hyperbolic Functions, the formulas are 
cosh(u v) = cosh u cosh v sinh u sinh v, 

sinh(u v) = sinh u cosh v cosh u sinh v ; 
112 



ADDITION THEOREM FOR ELLIPTIC FUNCTIONS. H3 

or, as they may be written, 

cosh( u v) = cosh u cosh v cosh u cosh v, 
sinh(i6 r) = sinh u sinhVsinh u sinh v ; 

and to these may be added 

,, v tanh it tanh -v 
tanh(u v) = -, ; 

1 tanh u tanh v 

constituting the Addition Theorem for the Hyperbolic Func 
tions. 

116. The Addition Theorem for the Elliptic Functions. 
For the Elliptic Functions the analogous formulas of the 
Addition Theorem are found to be 

sn(uv) = (sn u sn / t? sn w- sn v)/D, 
cn(u v) = ( en u en v + cn u cn v)/D, 
dn(u v) = (dn u dn v /c~ 2 dn% dnv)/D, 
where D=l /c 2 sn%sn 2 v ; 

or, performing the differentiations, and dropping the double signs, 
sn u en v dn v + en u dn u sn v /n 



cn u cn v sn u dn u sn v dn v 

1 K 2 sri 2 u sn 2 y 
dn it dn v /c 2 sn it cn u sn v cn v 



/ox 

(3) 



Putting /c = 0, we obtain the formulas for the Circular 
Functions, sin(u + v) and cos(w-f-i ), the denominator D re 
ducing to unity. 

Putting /c = l, remembering that then (16) snu becomes 
tanhu, cnu or dnu becomes sech u, we obtain from (1) 
tanh u sech 2 v + sech 2 ^ tanh v 



, 9 , 9 

1 tanh 2 w, tanh 2 y 

_ tanh u( 1 tanh 2 1?) + ( 1 tanh %) tanh v _ tanh u + tanh v 
1 tanh 2 u tanh 2 i; 1 + tanh u tanh v 

as before; with the corresponding formula for sech(i6-f-v) 
or cosh(w + r), the formulas for the Hyperbolic Functions. 

117. To establish these formulas of the Addition Theorem 
for Elliptic Functions, let us employ the geometry invented 
by Jacobi (Crelle, Band 3 ; Gesammelte Werke, I, p. 279), at 
the same time interpreting the geometry in connexion with 
Pendulum Motion. 

G.E.F. H 



114 



THE ADDITION THEOEEM 



To do this, let us suppose that P f would be the position of 
P in fig. 2 at the time t, if it had started r seconds later, and 
puU- T = f; then (6) 

AN = AD stfntf, N D = AD cnV, N E=AE drfnt , etc. ; 
and we shall prove that PP f touches a fixed circle through B 
and B during the motion (fig. 13). 




For suppose that, in the small element of time dt, P has 
moved to an adjacent point p and P to p ; and let PP , pp 
intersect in R, so that R is ultimately the point of contact on 
the envelope of PP . 

Then since, by a property of the circle, PP cuts the circle 
APP at equal angles at P and P , 

PR _, Pp _ velocity of P _ I ND 
RP ~ Pp ~ velocity of P ~ V Ntf 

Now describe a circle with centre o on AE, passing through 
B and B , and touching PP at a point which we shall denote 
by R ; then 

= OB* + Oo*-Wo.ON- Bo* 
= Oo(OD+Do + oO- 



Similarly, 
so that 



PR 



PR 



FOR ELLIPTIC FUNCTIONS. 115 

and therefore R and R coincide ; and we have thus verified 
that PP touches at R the circle oR (using the notation oR to 
mean a circle of centre o, and radius oR). 

Putting Oo = a, and denoting the angles A OP, A OP by 9, 
, and ADQ, ADQ by 0, ^, then 

PR 2 = 2a . ND = 4aR cos ^ = 4aZ/c 2 cos 2 0, 
so that P R+RP = 2 /v /(a 

while P P = 2Z sin J(0 - ), 

and therefore sin J(0 / ) = ^/(a/Z)K(cos i/r-f- cos 0). 
Putting nt = u, nt = v, nr = uv = iv; then since (8) 
, sin J$ = /c sin = /c sn u, cos J$ = dn u ; 
sin JO = /csini/r = 
(0 ) snudnv 



V<z 
j = 
6 



, a constant. 



I /f(CQS \lr + COS 0) 

Putting t = 0, v = 0, and therefore u = n-T = w, we find 

Va_ snw _lcmv_ 11cn.w. 
I 1 + cnw smu \l + cn^ 
so that 

1 cn(u v)_suu dn v dn u sn v _ en v en ?& 
1 + cn(i6 v) en v + en u sn u dn v + dn u sn u 

one form of the Addition Theorem, which by algebraical trans 
formation can be reduced to one of the preceding forms of 116. 

118. Representing, as in 31, snu by s v cnu by c lt dnu by 
d lt and the corresponding functions of v by s 2 , c 2 ,.c? 2 ; then 

1 + cn(u v) 

so that l-cn(u-t.j = fe-c i ; 

.-v) (c. 



V 



x 
or cn(u v) = 

and changing the sign of v, 




another form of the Addition Equation. 

A . 1 cn(u v) fs,d 9 

Again 4 = -" 

1+cnO-v) 

t A_ 2 - 
" 



and, adding numerators and denominators (componendo), 



THE ADDITION THEOREM 



. 2 (C 
(u V) = -9- 





-L K 8^ 82 

the usual form (2) of the Addition Theorem for the en function. 
But, subtracting numerators and denominators (dividendo), 

Cn(u V) ~~ nT 1 -~|- 
_l_ s /_\2 + K 2, 

*+lf> 



and another form can be easily established in the same way, 



(Glaisher, Messenger of Mathematics, vol. x., p. 106 ; 
M. M. U. Wilkinson, Proc. London Math. Soc., vol. xiii., p. 109; 
Woolsey Johnson, Messenger of Mathematics, vol. xi., p. 138.) 

119. Expressed again in Legendre s trigonometrical form, 
with = am u, i//- = am v, y = am(u v), 



ja _ 1 cos y _ sin <^> A^ sin 
V ^ sin y cos i//- + cos 

V _ 1 + cos y _ sin A\/^ + sin 
a sin y cos \^ cos 

Therefore, eliminating Ax//-, 



= 2 cos + 2 cos \/r cos y, 
or cos = cos \^ cos y sin -0- sin yA0. 

Expressed in Jacobi s notation, since u = 

cn.(v + iv) = cii vciiw 
Changing v + ^v into u v, this becomes 

cn(u v) = en u en v + sn u sn v dn(i6 v), 
or cos y = cos (p cos \^ + sin <^> sin \k-Ay. 

Conversely, these relations, treating y as constant, lead to 
the differential relations du dv = 0, 
or cZ0/A0-^/A^ = 0, 

or l - 2 sin 2 - - d 2 l - K 2 sin 2 < = 0. 



FOR ELLIPTIC FUNCTIONS. 117 

Writing x for sin <p sin \/s, y for cos (p cos \fs, and m for Ay, 
then cos y = x /(m 2 /c 2 )//c ( 17); and the integral relation 
becomes y 4- mx = ^/(m 2 /c /2 )//c, 

leading to the differential equation, of Clairaut s form, 

y - xp = *J(j? - K Z )/K, 

denoting dy/dx by p ; this is the form of the differential 
equation when we change to these new variables x and y. 

120. We have begun in 117 by supposing the points P and 
P to oscillate on a circle with velocity due to the level of the 
horizontal line BDB f , cutting the circle in B and B (figs. 2, 13); 
but if they are performing complete revolutions with velocity 
due to the level of a horizontal line BB through D not cutting 
the circle, but lying above it (figs. 3, 14), a similar proof will 
show that PP touches a fixed circle having with the circle 
PP the common radical axis BB , the two circles not inter 
secting ; and the Landen point L ( 28) will be a limiting 
point of these two circles. 

But this motion of P and P in fig. 14 is imitated by the 
circulating motion of Q and Q on the circle AQ in fig. 13 ; so 
that QQ touches at T a fixed circle, centre c ; and the hori 
zontal line through E is the common radical axis of this circle 
and the circle CQ, the Landen point L being a limiting point : 
and thus the Addition Theorem for Elliptic Functions can be 
deduced from the motion of P and P in fig. 14, or of Q 
and Q in fig. 13, as given by Durege, Elliptische Functionen, X. 

For if in fig. 1 4 a circle is drawn with centre o and radius 
oR, such that BDB (fig. 3) is the common radical axis of this 
circle and of the circle AP, then, since the tangents to these 
circles from D are equal in length, 



and now, if the tangent to the inner circle at R cuts the outer 
circle in P and P , 



as in 117 ; and similarly RP 2 = 20o . ND ; so that 

PR _ /^ / J D_ velocity of P . 

RP V XJJ velocity^fP" 

and therefore PP will continue to touch the circle R, during 
the subsequent motion of P and P . 



118 THE ADDITION THEOREM 

Similarly, in fig. 13, QQ during the motion touches a fixed 
circle, centre c and radius cT ; and putting Oc = c, 



We notice, on reference to 28, that 
LQ 2 = 2LC .EN=2LC.EA drfnt = 4l\I 
so that LQ = LA dn nt ; 

and therefore -~ = -^, 

or XT 7 bisects the angle QLQ in fig. 13; while LR bisects the 
angle PLP in fig. 14 ; we may state this theorem geometrically, 
" the segments of a tangent to one circle, cut off by another 
circle, subtend equal angles at a limiting point of the two 
circles." 

Then, with the notation of 117, 

QT+ TQ = 2 </(cQ(A^+ A0), 
and Q Q = 2R sin(0 - \j/) = 2 K H sin(0 - \/r) ; 

so that, in Legendre s trigonometrical form, 



Putting \/r = 0, then = y ; so that 



\^)_ /csiny 1 Ay 
/csiny 



VR /csin(0 + \^) K sin y 1 + Ay 
= : *- : * = or - , 
c A\^ A0 1 Ay /csiny 

the product of the two equations being unity. 
Conversely, the relation 

sin(0 + \fs} = (7( A^ + A0), 
where C is an arbitrary constant, leads to the differential relation 



121. Taking the equations 

Ay _ /c 2 sin(</> 



sny ir < sny 

we find, on eliminating sin 0, 
2/c 2 cos sin i/r sin y = (1 + Ay) (A^ - A^) - ( 1 - Ay)(Ai/r + A^) 



A^ = AyAx/>" /c 2 cos ^> sin i/r sin y, 
or dn u = dn v dn it; /c 2 cn it sn v sn u ? , 

with it = 



FOR ELLIPTIC FUNCTIONS. 119 

By eliminating cos 0, 

2/c 2 sin cos \}s sin y = 2 A^ 2 AyA</>, 

Ai/r = A< Ay + /c 2 sin cos \/r sin y, 

or dn(u 10) = dn u dn 10 -f /c 2 sn % sn w cn(it w). 

Changing w into v, 

dn(u v) = dn u dn v -f /c 2 sn u sn v cn(i6 r), 
or Ay = A0 Ai/r + /c 2 sin sin i/r cos y. 

Writing a; for /c 2 sin sin i/^, ^/ for A^Ai//-, and 7)1, for cny, 
then y + mx = *J(ic 2 + /c 2 m 2 ), 

the integral relation of Clairaut s differential equation 

y - xp = *J( K 2 + K z p z ), 
which is therefore the transformation of 



when we change to these new variables x and y. 

Taking the two trigonometrical expressions from 119, 120, 
for the Addition Theorem, 

1 cos y _ sin Ai//- sin 1//-A0 1 Ay _ K 2 sin(< i//-) 

siny cosi/r+cos< siny 

we obtain, by subtraction and reduction, 

Ay cos cos \[sA<p cos 

sin y sin <p + sin \js 

dn(u v) cn(i6 v) dn u en v en u dn v 

sn(u v) sn u -f sn v 

the form of the Addition Theorem given by J. J. Thomson 
(Messenger of Mathematics, vol. IX., p. 53). 

122. With the notation of the elliptic functions, 

1 + dn(u v) _ Ac(sn u en v + sn v en u} 
K sn(i// v) dn v dn u 

1 du(u v) /c(sn u en v sn v en u) 

/c sn(it v) dn v + dn u 

Therefore, as before, with Glaisher s abbreviations, 
v) _ (cZ 2 cZ 1 )(s 1 c 2 - 
~ 



120 THE ADDITION THEOREM 

Similar algebraical reductions to those given above for 
cn(u v) will establish the formulas for du(u v) and dn(u+v), 
given by Glaisher (Messenger, X., p. 106), 

S l^-2 C 2 S 2^1 C 1 






+ /c V 2 c i c 2 1 - A V 

the last of form (3), 116. 

123. The Duplication, Triplication, etc., Formulas. 
Putting v = u in formulas (1), (2), (3) of 116, and writing 
s, c, d for sn u, en u, dn u, we find 



9 = 



Writing ^f, 0, D for sn2u, en 2u, dii 2u, we find 



1+1) "" c/ 2 



_ 

~ 



Putting u=pr, then fif=l, 0-0, !) = /; and 



Again, in 67, 

, (1 +Qsn(^ /c)cn(u, /c) _ 1 +/ /l-dn(2u,/c) 
dn(u,/c) /c Vl + dn(2u,AcT 

and 2w=(l+X)v, X = (l -/)/(!+/), 



which is called Landeris second transformation. 



FOR ELLIPTIC FUNCTIONS. 121 

Again, putting v = 2u, and making use of the above formulas, 
we shall find 

o n 3,,- 



1 - 6* V + 4(1+ *Vs 6 - 3/cV 
1 - sn 3u 1 + */l - 2s + 2/cV - /c 



1 + sn 3u 1 - s\ 1 + 2s - 2/c 2 s 3 - /c 
l-jcsn3u_ 1 + K8/1 - 2/cS + 2/cs 3 - /cV\ 2 . 



1 + K sn 3u 1 - #s \1 + 2/cs - 2/cs 3 - jc 
with similar expressions for en 3 it and dn 3tt, leading to 

+ 2/c /2 c + 2/c 2 c 3 + * 



l-dn3u 1- 



l+dn3w, 

i_j_9i/r/3 9 t - / r7 

I ** <-l/ -K Uj 



the algebraical work is left as an exercise for the student. 

124. Poristic Polygons ofPoncelet, with respect to two Circles. 

Starting from the point A in fig. 13, and drawing the 
successive tangents AQ V Q^o, Q^Q^ to the inner circle, 
centre c, from the points Q v Q 2 , Q 3 , ... on the circle CQ ; 
or starting from A in fig. 14, and drawing the tangents AP V 
PJPv P 2 P B , ... to the inner circle, centre o, from P p P 2 , P 3 , . . . 
on the circle OP ; then, if we denote the first angle ADQ l or 
AEP 1 by am w, it follows from this construction that 

ADQ 2 = AEP = am 2^(;, ADQ 3 = AEP S = am 3w, . . . ; 
and we have thus a geometrical construction for the elliptic 
functions of the duplicated, triplicated, ... argument. 

When iv is an aliquot part, one 71 th , of the half period 27T, or 
T of the half period 2T seconds, then after n such operations 
the polygon AQ^^Q^, ... , or AP^^P^, ... , will close on itself 
at the starting point A ; and the preceding investigations show 
that during the subsequent motion of these points, the polygon 
formed by them will continue to be a closed polygon, inscribed 
in the circle CQ and circumscribed to the circle cT, or inscribed 
in the circle OP and circumscribed to the circle oR ; and thus 
we have a mechanical proof of Poncelet s Poristic Theorem for 
two circles, a problem discussed by Fuss, Steiner, Jacobi, 
Richelot, and Minding. 

(Cayley, Philosophical Magazine, 1853, 1854, 1861.) 



122 THE ADDITION THEOKEM 

Let us consider the particular cases of w equal to , J, J, }, * 
... of the half period 2K. 

(i.) When w = 2K, PP is horizontal in fig. 13; and P and 
P coincide in fig. 14. 

(ii.) When w = K, the circle oR in fig. 14 and the circle cT in 
fig. 13 shrink up into the limiting point L, Landen s point 
( 28) ; and now any straight line through L will divide these 
circles OP or CQ into two parts described in equal times, ^T ; 
while in fig. 13 the line PP will touch the circle described 
with centre E through B, L, and B , subtending an angle 4a 
at ; and any arc PP will be described in time \T, half the 
time of describing BAB ; hence the following theorem 

" Two segments of circles are described on the under side of 
the same horizontal straight line, one subtending twice as 
many degrees at the centre as the other; if a particle oscillates 
on the lower segmental arc under gravity, any tangent to the 
upper arc will cut off from the lower an arc described in half 
the time of oscillation." (Maxwell, Math. Tripos, 1866.) 

As P is passing through A in fig. 15, P is instantaneously 
at rest at B or jB ; and AB, AB are obviously tangents at B 
and B to the circle BLB , drawn with centre E ; while PP is 
one side of a crossed quadrilateral, escribed to this circle BLB t 
and inscribed in the circle BAB . 

When the circle cT shrinks up into the limiting point L, 
then, as in 120, 

QL 2 = 2CL. EN, LQ 2 = 2CL. EN 

and since QL . LQ is constant in the circle CQ, therefore 
EN. EN is constant, and equal to LE 2 , the value it assumes 
when N and N pass each other at the point L. 

Since EN . EN = EL 2 = EB 2 , 

a circle can be drawn passing through N, N , and touching EB 
at B ; and the triangles ENB, EBN are therefore similar, so 
that ENB = EBN , EN B = EBN. 

(Landen, Phil. Trans., 1771, p. 308.) 
Translated into a theorem of elliptic functions, 

EN. EN = EA 2 du 2 udu 2 v, and EB 2 = K 2 . EA 2 , 
so that, as in (59), 57, 

dn u dn v = K, when u v = K, 



FOR ELLIPTIC FUNCTIONS. 
Otherwise, since ( 28) 



and 
therefore 



123 



= ALdiiv, 
QL.LQ = AL.LD, 
dn u dn v = LD/A L = K. 




A 
Fig. 15. 

The similarity of the triangles A QL, LDQ shows that 



and since ( 10) AQ = ADsn u, DQ = 
therefore, as in (57), 57, 

sn u = en v/dn. v or cd v, when u> = v 
Again, since DQ /DL = A Q/LQ, 



therefore 



DL sn u 



AL dn u dn u 
as in (58), 57, when v=-u K. 



124 THE ADDITION THEOREM 

Conversely, if the straight line QLQ , passing through L, 
moves into the adjacent position qLq , then 



u <lQ _ Q^_ _ I EN _ velocity of Q 
q Q -LQ -^~EN ~ velocity of Q 7 

if Q and Q move under gravity, or diluted gravity, on the 
circle CQ with velocity due to the level of E\ so that QLQ 
will continue to pass through L, and will divide the circle CQ 
into two parts described in the same time \T ( 28). 

If in fig. 13 we denote the radius of the circle cT by r, then 



y or am w denoting the angle ADQ-^ ; while, from 120, 
1 Ay_ c . _R c, 

I+Ay""l > Cr y 



A , 2 4cR 2 (-R-c) 2 -r 2 

and thence J = * = 



Again, if Z)g is drawn from D to touch the circle cT, and 
the angle ADq is denoted by y or am w , then 

, r cosy , cnw 

sin y = -^ = / r- L , or sn ^v = -= -- , 

^L c Ay on 10 

so that ( 57) 



125. Poristic Triangles. 

(iii.) When w = \K or |-^T, triangles Q^Q^Qs can be inscribed 
in the circle CQ and circumscribed to the circle cT, while at the 
same time triangles P-,P 2 P 3 (or hexagons) can be inscribed in 
the circle OP and escribed to the circle oR (fig. 16). 
The well known relations of Trigonometry 

c 2 = R*- 2Rr, or a 2 = R 2 + 2Rr, 

where Cc = c, Oo = a, cT=r, oR = r, are now easily deduced. 
We may write these relations, more symmetrically, 
r r _ - 

^ ~ r 



In fig. 16, 

and since cQg bisects the angle J\T|Qj-4., which is equal to y, 
therefore DcOi = J (TT y) ; and DcQ| = DQ^c, or D(^ = DC. 

Similarly AQ = Ac f so that 



Therefore sin y + cos y = 1 , 

or sn \K + CD.$K = I , 

r r 

+ ~ 



FOR ELLIPTIC FUNCTIONS. 



125 



We shall employ this suffix notation for the points N, P, Q 
to signify points corresponding to aliquot parts of K. 

Corresponding to w = ^K, the circle oR becomes the circle 
through B, N^ R ; and nowP^AP* is a triangle escribed to 
this circle, and inscribed in the circle OP. 

For iv = ^K, the circle oR becomes the circle through 
B, N^ B ; and now we shall find that hexagons can be 
escribed to this circle, and inscribed in the circle OP. 




The tangents at P g , P* touch the circle BNJB , and the 
tangents at PI, Pj. touch the circle BNJ1 ; while AP%, AP< 
are the common tangents of the circles BNB f , BN 2 B . 

Denoting the sides of the triangle Q^Q^ by q v q z , g 3 , then 



But u v i// 2 , M 3 denoting the value of u corresponding to the 
points Q v Q 2 , Q B , and cZ 1? cZ 2 , d B denoting the corresponding 
values of dn u, then ( 120) 



(r7 2 



so that 



a constant, a relation connecting d v d 2 , d s , when 



126 THE ADDITION THEOREM 

126. Poristic Quadrilaterals. 

(iv.) When w = ^K, quadrilaterals Q^Q^Q^Q^ can be inscribed 
in the circle CQ which are circumscribed to the circle cT, and 
now the corresponding relation is found to be 



( -< v 

\Ji-J 



while TjjPg, T 2 T intersect at right angles in L, being the 
bisectors of the angles between Q^LQ^ Q?LQ 3 (fig. 17). 

This relation is proved immediately by taking the quadri 
lateral in the position A QDRs ; and now y = y = am ^K, 



so that squaring and adding leads to the desired relation. 

As in (ii.), quadrilaterals can be escribed to the circle BLB , 
which are inscribed in the circle OP, since N coincides with L. 

But the circles BNiB and BN*B are related to the circle 
OP with regard to poristic octagons; and the common 
tangents of these circles are easily recognised at the points 

PJ, p 4 , PJ. 

Conversely, starting with the circle cT and the internal 
point L, and drawing T^LT^ TJLT through L at right angles 
to each other, the tangents to the circle cT at T v T 2 , T 3 , T 4 
will form a quadrilateral Q 1 Q 2 Q 8 Q 4 which is inscribed in a 
circle CQ, the diagonals Q^, Q 2 Q 4 passing through L, and 
being equally inclined to T^T 3 and jT 2 T 4 . 

If Q-fi, Q 2 c, Q^c, Q 4 c are produced to meet the circle CQ again 
in q v q z , q 3 , q, then q-^q 3 and g 2 g 4 are diameters of the circle 
CQ; for Q^ bisects the angle Q 2 QiQ 4 > so that the arc 
Q 2 g 1 = arc q-tQi, and similarly the arc Q 2 ^ 3 = arc g 3 Q 4 , so that the 
arc q l Q 2 q 3 = &YC yiQ$3> an( i eac ^ ^ s therefore a semi-circle. 

It follows, from elementary geometrical considerations, that 

LT* + LT* +LT 3 *+ LT? = 4r 2 , 
or T^ 

, 



so that cq* + cg 3 2 = cq* + cg 4 2 = (^ 2 - c 2 ) 2 /r 2 , 
leading to 2 (R 2 + c 2 ) = (^ 2 - c 2 ) 2 /r 2 , 

or, as before, +-J= 1. 



FOR ELLIPTIC FUNCTIONS. 127 

t> Denoting by u v u 2 , u 3 , u^ the values of u at Q lt Q 2t Q 3 , Q 4 , 
so that u^ - u. 2 = u 2 - u 3 = u 3 - v, 4 = \ K ; 

and denoting by cZ lf c^ 2 , cZ 3 , cZ 4 the coiTesponding values of dn u, 
then ( 5 7) d 1 d 3 = d 2 d^ = K ; 

and ( 120) ZQ = 2Z(l-/ c / )dnu, 

so that " " ^ " 

while 



= 21(1 - K )(d^) ; 




Fig. 17. 
Now by a property of the circle (Euclid VI. D) 

so that 



- ^^(^i + d s )(d 2 + cZ 4 



or (^i + ^ 3 )(^ 2 + cZ 4 ) is constant, and =2 N //c / (l+ 
the value obtained by putting u 4 = 0, when 



and d* = 



128 THE ADDITION THEOREM 

Then 

when 
Thus 



so that 



127. Poristic Pentagons, etc. 

(v.) When v = \K, or -fTf, the poristic polygons are pentagons 
(fig. 18), and the relation to be satisfied is of the form 



or p-q=p + q-i 

where p and q are used to denote r/(R c) and r/(E + c). 

We notice that the relation for pentagons leads to a cubic 
equation, when two of the three quantities R, r, c are given ; 
but the equation reduces to a quadratic when c = or the circles 
are concentric, the case considered by Euclid. 

The reader is referred to the articles of Cayley (Phil. Mag., 
Series IV., Vol. 7, and Collected Works) and to Halphen s 
Fonctions Elliptiques, t. II, chap. X., for the proof of this 
relation and the similar relations for other polygons. 

We shall find that Halphen s a and y (t. II, p. 375) are con 
nected with our R, T, c, /c, and w by the relations 



v> -r -- 

(R + c) 2 r 2 * \R + c 

and thence Halphen s x and y can be formed. 

By the use of Legendre s Table IX. for F(<p, K ) (F. E., t. II.) 
we are able to construct geometrically, to any required degree 
of accuracy, figures of circles related to each other for poristic 
polygons of any given number n of sides. 

Having selected an arbitrary modulus K or modular angle 
Ja, we look out the value of K, and then determine, by pro 
portional parts, the value of in degrees corresponding to an 



FOR ELLIPTIC FUNCTIONS. 



129 
will mark 



amplitude of K/n, 2K/n, . . . ; and these values of 
the position of the points Q v Q 2 , .... 

Thus, in drawing figs. 13, 14, 16, 17, we have selected 
/c = sin 60, when K= 2 1565; and in drawing fig. 16 for poristic 
triangles, we find, from Legendre s Table IX., 

am J#=c.m. of 3S49 , amf^T=c.ir,. of 685 . 




A 
Fig. 18. 

These angles enable us also to set out figs. 13 and 14, where 

the circles are drawn so related as to admit of poristic hexagons. 

In drawing figs. 15 and 17, Landen s point L is sufficient to 

complete the diagram ; also to double the number of sides of 

a polygon of an odd number of sides. 

In fig. 18, K has been taken as sin 75, as in figs. 1, 2, 3 ; and 
now K= 276806 ; and from Legendre s Table IX., 

amifiT=c.m. of 3018 , am|/i=c.m. of 7020 / , 
by means of which the figures can be drawn. 

Fig. 19 shows poristic heptagons, to the same modular angle 
of 75, laid out by means of the relations 

0! = am 4^=0.001. of 22S , 3 = am4A r =c.m. of 5649 , 
5 = amffiT=c.m. of 776 . 

G.E.F. 1 



130 



THE ADDITION THEOREM 




128. The poristic relation between the quantities R, r, c 
has been obtained by placing the polygon in a symmetrical 
position; but another method is employed by Wolstenholme 
(Proceedings London Math. Society, vol. VIIL, p. 136 ; also 
by Halphen, F.E.. II., chap. X.), where the polygon on the circle 
OP is considered in its limiting form, when passing through 
one or both of the common points B and B . 

Thus with triangles, the tangent to the circle oR at B must 
meet the circle OP again at a point PI, the point of contact of 
a common tangent of the two circles P and R, the degenerate 
triangle being BPP. 

For quadrilaterals, the tangents to R at B, B must meet at 
A on the circle P, BACAB being the degenerate quadrilateral. 

For pentagons we obtain the degenerate form BP^Pg_P t PB, 
where BP^ is the tangent at B to oR, the circle through 
B, JV|, B , and PT. is the point of contact of a common tangent 
of the circles OP and oR (fig. 18). 

For hexagons (fig. 16) the limiting form is BP^P^BP.P^B, 
where BP^, P^B f are tangents at B, B to the circle through 
B, JVi, B r : and so on. 



FOR ELLIPTIC FUNCTIONS. 131 

129. Geometrical Applications of Elliptic Functions to 
Spherical Trigonometry. 

Taking the fundamental formulas of Spherical Trigonometry 
cos c = cos a cos b + sin a sin b cos (7, 

sin A sin B sin C 

= . -= - = K , suppose; 
sin a sin 6 sm c 



then cos C = ^/(l /c 2 sin 2 c) = Ac, 

so that cos c = cos a cos b -f- sin a sin 6 Ac, 

a formula like that of 119, with a, b, c for 0, ^, y ; so that if, 

keeping (7, c, and therefore K constant, we vary a and b, then 

cos B . da + cos A .db = 0, 
or dafAa db/Ab = ; 

and, conversely, the integral of this differential relation is the 
formula above. 

(Lagrange, Theorie des fonctions, p. 85, 81, 82 ; 
Legendre, Fonctions elliptiques, t. I., p. 20.) 
If, in Jacobi s notation, we put 

a = am(u, K), b = am(r, /c), c = am(vj, /c), 
then the differential relation becomes 

du dv = 0, 

so that u v = & constant = w, 

since a = c, or ^t = 10, when & = and v = 0. 

Supposing K is less than unity, and the angle C is acute, then 
c>C, and of the other angles, one, A, must be obtuse, and the 
other, B, acute. 

But by changing to the colunar triangle on the side BC, we 
may convert the triangle ABC into one in which all three 
angles are obtuse ; and in such a triangle we may put 

a = am u, b = jr am v = am(2 J fiT v), c = am(2^T w) ; 
so that if the triangle ABC has three obtuse angles, we may put 



s , 



where it 1 -f- u. 2 + U B = u + 2K v + 2K lu = 

and now 

cos A = dn u v cos B = dn u. 2f cos C = dn u, 
so that, by 29, we may write 
A = TT amC/citp I/AC), B = -TT am(/at 2 , l//c), C=TT am(/cW 3 , l//c), 
where /c is less than unity. 



132 THE ADDITION THEOREM 

For instance, if ABC is the spherical triangle formed by three 
summits of a regular tetrahedron, 

A = B = C = ITT, 
and cos a = cos b = cos c = J, 

sin a = sin b = sin c = f >^/2, 

sin a 4^2 8 * 8 * ** ~ 16 
while ^ = ^2 = ^3 = j^K, 

so that en K= - }, sn |JJT= f */2, dn 4^= f . 

When /c = 0, K=\-w, and the triangle J.5(7 is coincident with 
a great circle ; and now 

When K = 1, K=vo; and therefore of u v u 2 , U 3 , two of them, 
say i&j and i& 2 , are infinite ; so that 

cos a = sech u x = 0, or a = JTT ; and similarly b = JTT ; 
the triangle ABG now has two quadrantal sides and therefore 
two right angles, the third side c and angle G being equal, and 
taken greater than a right angle. 

130. For values of K which would be greater than unity, we 
change the notation by considering the polar triangle; and now 
if ABO is such a polar triangle, having three acute sides, instead 
of three obtuse angles, we put 

sin a _ sin b _ sin c _ 
sin A sin B sin C 

and A = am v lt B = am v 2 , 0= am V B , 

where v l = 2Ku l , v 2 = 2K u- 2 , v% = 2K u s , 

so that ^ + ^ + ^ = 2^. 

Now sin a = K sn v v sin b = K sn v 2 , sin c = K sn v 3 ; 
cos a = dn v v cos b = dn i> 2 , cos c= dn V 3 ; 
so that ct = am(/c^ 1 , l//c), & = am(/ci> 2 , l//c), c = am(/c^3, l//c). 
The fundamental formula 

cos c = cos a cos b + sin a sin b cos c 
now leads to the formula of 121, 

dn V B = dn ^dn v 2 +/c 2 sn ^sn u, en v s , 
or dn^ + v 2 ) = dn Vjdn v 2 /c 2 sn ^sn ^c^Vj + V 2 ). 

In the degenerate case of /c = 0, K=^-JT ) and 

and now a = 0, 6 = 0, c = 0, so that the spherical triangle is 



FOR ELLIPTIC FUNCTIONS. 133 

indefinitely small, and may be considered a plane triangle; 
and we can thus deduce the formulas of Plane Trigonometry. 

131. A spherical triangle thus falls into one of two Classes, 
I. or II. ; in Class I. the triangle, or a colunar triangle, has 
three obtuse angles; in Class II. the triangle, or a colunar 
triangle, has three acute sides ; the quadrantal triangle falling 
into Class I., and the right-angled triangle into Class II. 
In Class I. we put 

sin -4 _sin jB_sin C _ 
sin a ~ sin 6 sin c ~ 
and then K is less than unity; and we put 

a = am u l} b = am u.- c = am u s , 
where u l + u 2 + u 3 = K, 

and then 

A = TT Sim(KU v l//c), B = TT am(/cu 2 , l//c), C=7r am(/cU 3 , I/*). 
In Class II. we put 

sin a _ sin b _ sin c _ 
sin A ~ sin B ~ sin G~ 

and then K is less than unity ; and we put 

A = amv 1 , .Z? = amt 2 , 0=ami i s , 
wh ere v l + v 2 + v 3 = 2K, 

and then a = am(/cf 1 , l//c), 6 = am(/ci 2> l//c), c = am(/ci 3 , l//c). 

When this triangle of Class II. is the polar of the triangle 
in Class I , u x + 1\ = u. 2 + v. 2 = u 3 + v 3 = 2K. 

The change from one Class to the other affords an illustration 
of the change from one modulus to the reciprocal modulus ( 29). 

The spherical triangles employed originally by Lagrange 
and Legendre fall into Class I.; and a full discussion of the 
connexion between Elliptic Functions and Spherical Trigono 
metry will be found in the Quarterly Journal of Mathematics, 
vols. 17, 18, 19, in articles by Glaisher and Woolsey Johnson. 

But it is preferable in some respects to work with the 
spherical triangles of Class II., as growing out on the sphere 
more naturally from the infinitesimal plane triangle ; so it is 
proposed to develop here the relations with Elliptic Functions 
by means of a typical triaDgle of Class II., having three acute 
sides, and to refer to the articles of Glaisher and Woolsey 
Johnson for the corresponding relations of Class I. 



134 THE ADDITION THEOREM 

132, Writing c v s v cZ x for cnv l} snv v duv v etc. : then with 

v l +v 2 +v 3 = 2K > 
we may put, in Class II., 

A=a,mv l , B = &mVz, (7=amv 3 ; 
so that cos A = c v sin A = s v etc. ; 

and now sin a = K sin A = KS I} cos a = d v etc. 
From the fundamental formulas 

cos c = cos a cos b + sin a sin b cos C, 
cos 0= cos A cos 5 sin A sin 5 cos c, 
we obtain d = d 



where <i 3 = dn i; 3 = dn(^ 1 + v 2 ), c 3 = en v 3 = cn(v 1 + 1 2 ) . 

Again, from these two formulas of spherical trigonometry, 
cos (7= cos A cos B sin jl sin .B(cos a cos 6 -f- sin a sin 6 cos (7), 
o ^ _ cos A cos 5 sin A sin 5 cos a cos b 

- COS L/ --- - - ; - - - ; - - ; - - ; - - - - 

1 sin A sm B sin a sin 6 
so that -CD- 



. ., -, cos a cos 6 sin a sin 6 cos A cosB 

Similarly, cosc = = : ^ ; --- : 5 - -, 
1 sm J. sin B sm a sin 6 

leadin to d = dv 



12 

As a specimen of Class II., take the spherical triangle formed 
by three adjacent summits of a regular icosahedron ; then 
A=B=C=fr; 

, cos C+ cos A cos B cos (7 1 

and cosc = : : ^ -- = T~ n n=> 
sin J. sm B 1 cos u ^75 

so that K = sine/sin C = 1^7(10 2^/5); 

and then ^ = ^2 = ^3 = !^, 

so that en K =cosC= K^o - 1 ), 

dn f K = cos c = -J-^/5. 

133. To prove that in a triangle of Class II. we obtain the 
differential relation 

cos b. dA + cos b. dB = 0, or dA/AA + dB/AB = 0, 
when we change A and 5, keeping c and (7 constant, dis 
place the triangle ABC into the consecutive position ABC\ 
keeping the points A, B fixed and the angle AC S unchanged 
in magnitude (fig. 20). 



FOR ELLIPTIC FUNCTIONS. 



135 



Then, if CA and CB produced on the sphere meet the great 
circle of which C is the pole in P and Q, the arc PQ = G ; and 
if C A and C B produced meet this great circle in P f and Q , 
the arc P Q is ultimately equal to the arc PQ, or 




Fig. 20. Fig. 21. 

But PAP = -dA, QBQ = dB; while ultimately 

PP = -sin 4P . cZJ. = -cos 6 . dA, QQ = cos a . dB; 
so that cos b . dA + cos a . dB = 0, 

or 
since sin a = KsinA, 

With A = am v v B = am v z , this becomes 



so that v x + v. 2 = constant 2K v^ where C = am v s ; 
since B+C=7r, or v. 2 + v% = 2K, when A = 0, i\ = 0. 

Conversely, this differential relation, interpreted with respect 
to the triangle ABC, of which the side AB is fixed, expresses 
the constancy of the opposite angle C. 

134. If, as is customary, we deduce the differential relation 

cos B . da + cos A . db = 0, or da/Aa + db/Ab = 0, 
from a spherical triangle ABC of Class I., in which 

sin A=Ksiua, cosJ. = Aa, 

we keep the angle C fixed, and displace the side AB into its 
consecutive position A B , without change of length, through 
an infinitesimal angle 6 about the centre of instantaneous 
rotation /, the point of intersection of the arcs AI, BI, drawn 
perpendicular to CA, CB respectively (fig. 21). 

sin IBH cos B 



m , db ,, A A sm/J. 

Then --=11-=-. - 

da BB sin IB 



sin I AH 



cos A 



136 THE ADDITION THEOREM 

135. To obtain immediately the addition formulas (1), (2), 
(3) of 116 for the elliptic functions, Mr. Kummell draws the 
arc CD perpendicular to AB (fig. 20), and denotes the perpendi 
cular CD by p, the segments BCD, ACD of the angle C by 
F, G, and the segments BD, DA of the base C by / g ; so that 
F+G=C t f+g=c. 

(Kummell, Analyst, vol. V., 1878.) 

Now, from the right-angled spherical triangles ACD, BCD, 
cos G = sin A cos fr/cos p, sin G = cos A /cos p ; 
cos F= sin B cos a/cos p, sin F= cos J9/cos _p ; 
or with sin A = s v cos J. = c 1? sin a = ATS, cos a = d v etc., 
and writing M for cos p, 

cos G = s^dJM, sin G = cJM ; 
eosF=s 2 d l /M, siuF=c 2 /M. 
Also sin 2? = sin A sin 6 = sin a sin 5 = KS^, 

so that ^f 2 = cos 2 p = 1 - K \ V, 

a quantity which we have found it convenient to denote by D. 

Now, cos C= cos F cos 6r sin .F sin G, 

or c 3 - (s 1 s 2 cZ 1 cZ 2 - c^yD, 

or en (v + v 2 ) = en v 3 = (c t c 2 s 1 s 2 c? 1 cZ 2 )/D, 

formula (2). 

Again, sin Csii\(F-\- G) 

= sin jp 7 cos G 4- cos J^ 7 sin G, 
or s 3 = (s^gdg + sfrdJ/D, 

where 8 ft =8HV 8 =3Sn(^ 1 -|-V 2 ), as in formula (1). 

Changing the sign of v 2 , 

sn^-^) =sin( J F- G), 
or J^ 7 6r = am(v 1 v 9 ), 

while ^+(7 = am^ 3 = am(2^r t^ -y 2 ) 

= 7r-zm(v l + v 2 ), 

so that jP== JTT \ am( 

G = ATT - J am 
Thus, for instance, 

tan{ | am(^ 1 + v 2 ) + J am^ v 2 )} = cot (r = tan A cos 6 = s^d^jc^ 
tan { | am(i; 1 + v z ) J am(^ u>)} = cot J^= tan B cos a = s^djc^ 
Again, from the right-angled spherical triangles BCD, ACD, 
cos /= cos a/cos p = dJM, sin /= sin a cos B /cos p = KS^/M ; 
cos $ = cos 6/cos p = cZ 2 /Jf, sin g = sin 6 cos A /cos p = 



FOR ELLIPTIC FUNCTIONS. 137 



and therefore 

dn(i\ + v. 2 ) = dn v s = cos c = cos(/+ g) 
= cos/cos g sin/sin g 



_d l d 2 



as before, in (3), 116. 

Also sin(/+0) = /csn(v 1 + v 2 ), sin (/-</) = K sn(v l -v 2 }-, 
whence /and g can be found as functions of v^ + v^ and i\ v z . 

136. The formula employed by Morgan Jenkins in the 
Messenger of Mathematics, vol. XVIL, p. 30, as fundamental 
in Spherical Trigonometry, is 

sm(A+B) _ sinO 
cos 6 + cos a 1 + cosc " 
and this now leads to 

^1^9 ~r ^2^1 _ ^3 
~ 



or, in the Legendrian form 

sin G 



1 + A(7 

a formula already obtained from pendulum motion in 120. 
Then the formula 

S 1 C 2 ~~ S 2 C l S 3 

d^-d^ l-cZ 3 

or sin( J. B) _ sin G 



AB-AA 1-AO 5 
gives sinU-J)_ sinO 

cos 6 cos a 1 cos c 
The formulas of 120, in the form 



1 -f c 3 c 2 c : 
lead to the relations 



sin(q + fr) _ sine 

C" ..................... 





cos 5 + cos J. ~ 1 -cos C" 
sin(q b) sin c 



cos B cos J. "~ 1 + cos (7 

and from these four formulas of Spherical Trigonometry Mr. 
Morgan Jenkins deduces the analogies of Xapier, Delambre, 
and Gauss. 



138 THE ADDITION THEOREM 

137. Write, as before, in 135, 
A = am u, B = am v. 
F= \-K \ am(u + ^) + J am(u, / y), 
6r = JTT J am(M- -j- i>) J am(u -y). 
Then, since 

siu(F 4- ) + 8in(F- G) = 2 sin .Fcos G, 

therefore, writing c v s v d v for en u, sn u, dn u, and c 2 , s 2 , cZ 2 for 
en -y, sn v, dn v, and D for cos 2 ^ or 1 K\\ 2 , 

sn(u + f) + sn(u v) = 2s l c 2 d 2 /I) ) ............ , ...... (1) 

cos(.F- G) - cos(,P+ G) = 2 sin J^sin G, 
cn(u v)+ cn(u + v) = 2 c^cJD ; ..................... (2) 

cos(/ (/)+ cos(f+g) = 2 cos/cos gr, 

dn(u + -y) = 2 d^dJD ; ..................... (3) 

in(^- G) = 2 cos ^ sin G, 

sin(u v) = 2 s^dJD ; .\ ............ ..... (4) 

cos(,P- G) + cos(T+ G) - 2 cos 7^ cos G, 
cn(u v) cn(u + v) = 2 s^s^dJD ; ................. (5) 

cos(f-g) cos(f+g) = 2sinfsing, 

dn(u + v) = 2 K \c 1 s 2 c 2 ID , ............... (6) 

sin(F- ^-sin^-sin 2 ^, 

sn(u - v) = (c 2 2 - c^VD = (s^ - s 2 2 )/D. ..(7) 
Again, since 

1 + sin(/+ <7)sin(/-gr) = cos 2 ^ + sin 2 /, 
and sin(/+^) = /csn(u + ?;), sin(/ #) = * sn(i6 -y), 

-7;) = (^ 2 2 + /cVc 2 2 )/Z); ............ (8) 

- G) = sin 2 ,P+ cos 2 G, 
-v) = (c 2 2 + s 1 2 ^ 2 2 )/D; .............. (9) 

1 - coa(F + G)cos(F- (7) = sin 2 G + sin 2 ^, 

1+ cn(u + v) cu(u-v) = (c 1 * + c*)/D ;..... .......... (10) 

1+ cos(f+g) cos(/ r/) = cos 2 /cos 2 ^, 

(11) 



-/c 2 sn(u + v) sn(tt -v) = (^ 2 + Ac 2 s 2 V)/D ; .......... (12) 

- sin(.F+ )sin(^- G) = siri 2 G + cos 2 F, 

(13) 



; ........ ..(14) 

cos(/ g) = siu 2 f+sm 2 g, 

: , ........ (15) 



FOR ELLIPTIC FUNCTIONS. 139 

{1 sin(.F+ G)}{lsm(F- G)} = (sin ^cos ) 2 , 



(16) 
(17) 
............. (18) 

............. (19) 

8(^4- G)}{lcos(F- G)} = (sin Fsin G) 2 , 

(20) 



{1+ cn(u + v)}{! cn(u-v)} = (8 l d z +8 z d. 1 )*/D; ............ (21) 

(1 cos(/+0)}{l cos(/-(/)} = (cos/cos) 2 , 

-v)} = (d l dJ*ID; ................ (22) 



dv(u-v)}=K 2 (s l c< i +s 2 c l ) z /D; ........... (23) 

sin(F + 6r)cos(jF G} = sin G cos G + sin F cos F, 
sn (u + v) cn(i6 v) = (s&d^ + s^d^/D ; ......... (24) 

- sin(F- G)cos(F+ G) = sin G cos G - sin .Pcos F, 
sn(w v) cn(i6 + v) = (s&dz s&dJ/D ; ...... , ..(25) 

sin(/+ g) cos(f-g) = sin/ cos /+ sin g cos g, 
sn(u + v) dn(u v) = (s^c, + 8 2 d z c^)/D ; ......... (26) 

siv(f-g) cos( f+g) = sin /cos/- sin g cos g, 
sn(u v) dn (u + v) = (s-^cl^ s^d^sJD ; ......... (27) 

-cos(.F+ G)cos(f-g) = {cos A cosB-sinAsiuBcos(f+g)}cos(f-g), 
cn(u + v) dn(u - v) = (c^d^l 2 - K\s^jD ; ....... (28) 

cos(F- G) cos (f+g) = cos(F-G){ cos a cos b + sin a sin b cos(F+ G)}, 
cn(u v) dn(u + v) = (c^c. 2 d^d 2 + K ^S^/D ; ...... (29) 

eon 20= 2 sin 3 eos 0, 
sin{am(u + v) + &m(u v) } = 2 s^d^/D ; ................... (30) 



) am (u v)} =28 2 c z dJD ; .................. (31) 



cos{am(u + v) + am(u - v)} = (q 2 - sfd^/D ; ............ (32) 

- cos 2F= sii^F- cos 2 ^, 
cos{am(^ + r)-am(u-t )} = (c 2 2 -s 2 2 c? 1 2 )/j[); ............. (33) 

the thirty-three formulas of Jacobi, given in his Fundamenta 
Nova, 18, and reproduced in Cayley s Elliptic Functions. 



140 THE AUDITION THEOREM 

Similarly any other formula in Spherical Trigonometry is 
converted into a form of the Addition Theorem of the Elliptic 
Functions, and conversely; by writing c v s-^ for cos A, sin A, 
and d v KS I for cos a, sin a, etc., with 



Thus the six four-part formulas, of which 

cot a sin c = cot A sin B+ cos c cos B 
is the type, obtained by eliminating cos b between (a) and (J3), 

lead to Sg^ = s 2 c i + S i c 2^3> 

with five other similar relations. 

By means of these and the preceding relations we can prove 
the following examples on the formulas of Elliptic Functions. 

EXAMPLES. 

1. Prove that, if u-\-v + w+x = 0, 

/. N en u dn v dn u en v , en w dn x dn w en x _ ~ 
snu suv snw s 



(ii.) /c 2 /cV 2 sn USUVSUWSD.X + /c 2 cn u en v en w en a? 
dn u dn t> dn w dn a; = 0. 

2. Prove that 

,. x , N x 2/c 2 sn u en i> dn v 

(I.) nsftt tn-f-8n( < tt+tj)= :r~9 -- ;r~ ? -- > 

dn 2 ^ dn 2 i6 

(ii.) 1 - K 2 sn 2 O + v)sn 2 (u, - v) = (1 - /c 2 sn%)(l - /c 2 sn 4 ?;)/D 2 : 
(iii.) /c 2 sn(u + v)sn(u v)su(u + ^)sn(u w) 



- K 2 sn 2 v sn 2 ^(;) 
~ 



2 



sn 2 (u 

l-snu_cn 2 i( 




... 
(u.) 



4. Prove that 



and hence prove that the expression 

1 K sn x sn y 1 + K sn z sn w 
1 + K sn x sn y 1 K sn z sn w 



FOR ELLIPTIC FUNCTIONS. 141 

remains unaltered when for x, y, z, w we substitute respectively 



5. Prove that, if tanh A = K sn 2 a, tanh B = K sn 2 /3, 

tanh(4-5)=*sn 
Deduce Jacobi s relations, 



or 

+ y)sn(/3-y) 1 - /c sn(y + a)sn(y-g) l-/ 



y)sn(/3-y) l+/csn(y+a)sn(y- a ) 

or = 1 ; 

1 - K sn(t - x)sn(y - z) ~L - K $n(t - y)sn(z - x) l-KSu(t-z)sn(x-y} 

l+KSn(t-x)sn(y-z) 1 + K sn( - y)sn(z - x) 1 + K sn(t - z)sn(x - y) 

or =1 ; 



(Glaisher, Q. J. M., vol. XIX., p. 22.) 

6. Prove that the tangents at the points on an ellipse of 
excentricity e whose excentric angles are 

<p = JTT am(u, e), i/r = JTT am(i;, e), 

will meet on a confocal ellipse when u v is constant, and on 
a confocal hyperbola when u + v is constant. 
Hence show that the general integral of 

d<p/J(l - e 2 sin 2 0) - cty/JQ - e%in 2 ^) - 
may be written 



and convert this into the form 

cos y = cos < cos \[r+ sin sin ^-^/(l e 2 sin 2 y), 

proving that tan^V 



7. Prove that the straight line joining the points 

ccu(u-\-v) } csn(i6 + -u) and ccn(u v), csr\(u v), 
on a given circle of radius c, will touch an ellipse whose semi- 
axes are c sn(/f v), ccnv, when u is constant and v is 
variable ; and determine the envelope when u is variable and 
v is constant. 



CHAPTER V. 

THE ALGEBEAICAL FORM OF THE ADDITION 
THEOREM. 

138. The first demonstration of the existence of an Addition 
Theorem for Elliptic Functions is due to Euler 
(Acta Petropolitana, 1761 ; Institutiones Calculi Integralis), 
who showed that the differential relation 



connecting X = ax 4 + 4>bx s + Qcx 2 + 4?dx + e, 

or (a, b, c, d, e)(x, I) 4 , 

the most general quartic function of a variable x, and Y the 
same function of another variable y, leads to an algebraical 
relation between x and y, X and Y. 
This algebraical relation is 

_*JX\ 2 = a (x + y) 2 + M(x + y) + C, 

x y / 

where C is the arbitrary constant of integration ; and this 
relation when rationalized leads to a symmetrical quadri- 
quadric function of x and y, of the form ( 148) 

ax 2 y 2 + 2/3xy(x + y) + y(x 2 + xy + y 2 ) + 2% + y) + e = 0, 
or (ax 2 + 2/3x + y)?/ 2 + 2(/3x 2 + 2y + % + y^ 2 + 2&c + e - 0, 
or (ay 2 + 2/fy + y)x 2 + 2(/fy 2 + 2yy + S}x + y if + 2Sy + e = 0. 

(Cayley, Elliptic Functions, chap. XIV.) 
With a = and b = 0, X and Y reduce to quadratic functions 
of x and y ; and then 

*J X ~^Y =gi consfcant 

x-y 
is the general integral o 

142 



ALGEBRAICAL FORM OF ADDITION THEOREM. 143 

139. By writing (lx+m)l(l x f +m ) for x, which is called a 
linear substitution, this symmetrical quadri-quadric function 
becomes unsymmetrical, the five constants a, /3, y, S, 6 being 
thereby raised in number to nine ; and then 

dx/^/X becomes changed to (lm f I mjdx I^X , 
where X = (a, b, c, d, e)(lx + m, I x + m )*. 

The invariants g<> and g z of the quartic X have been defined 
in 75, and in 53 the discriminant A=# 2 3 27<7 3 2 , and the 
absolute invariant J=gf/A ; and now, if g 2 , g^ A , J denote 
the same invariants of X , we find 

g. 2 =(lm -l myg 2 , gj = (l m-lmjg A = (Zm - Z m) 12 A ; 
while the absolute invariants J and J are equal. 

Conversely, any unsyrnrnetrical quadri-quadric function 
whatever of x and y may be written 



G(x, y) = (ay* + 2/3 y + y> 2 + 2(/3if + Zy y + S")x + yif- + H y + e" 



L, M } N being quadratic functions of x, and P, Q, R being 
quadratic functions of y. 
Then by differentiation 

(Px + Q)dx + (Ly+iM)dy = ; 
and by solution of quadratic equations 

Ly + M= *J(M- - LN) = JX, suppose ; 
Px+Q = J(Q 2 -PR) = JY, suppose; 
and thus we are led to the differential relation 



where X and T are quartic functions of X, not necessarily of 
the same form, but having the same g. 2 and g 3 . 
A linear transformation, such as that given by 



can however always be found, which will transform 



where T is a quartic having the same coefficients as the quartic 
X ; in other words, the quartics X and Y have the same in 
variants ; so that we may, without loss of generality, consider 
X and Fas of the same form, and therefore drop the accents 
in the expression for G(x t y}. 



144 THE ALGEBRAICAL FORM 

Now 



so that *- = axy + p( 

x y 

a form of the integral relation, in which the coefficients a, 6, c, 
d, e in X and Y are functions of a, /3, y, S, e, determined by 



e), 
the Hessian, with changed sign, of (a, /3, y, (5, e)(x, I) 4 ; and 



140. Lagrange proves Euler s Addition Equation as follows: 
Put dx/dt^^/X, and therefore dy/dt= ^/Y] then 



suppose; so that putting x + y=p, x y = q, then 



dp dq == x_ Y 
dt dt 

= %apq(p 2 + q 2 ) + bq(3p* + g 2 ) + Gcpq + 4>dq ; 



whence 

dt 

2 dp d?p 2 dqfdp\ 2 c/p 

~ = 



Both sides of this equation are now integrable, so that 



or 



We notice here that, if C=4b 2 /a, 
X- 
x-y 



OF THE ADDITION THEOREM. 145 

141. In the canonical form considered by Legendre, with 
oc = suu, dx/du = ^/(l x 2 . 1 /c 2 ^ 2 ), 

y = sn v, dy/dv = ^/( 1 y 2 . 1 /c 2 2/ 2 ), 

then X = 1 x 2 . 1 A 2 , Y= ly 2 .! K 2 y 2 . 

Therefore dxj^/X + dy/J Y= 0, 

leads to du+ dv =0, 

or u+ v = constant; 

which, in Clifford s notation, may be written 

sn ~ l x -\- sn ~ l y = con s tant. 
Euler s Addition Theorem of 138 now gives 



_ (en u dn u en v dn v) 2 /c 2 (sn 2 & sn 2 i?) 2 

(sn u sn vf 
_ /dn u en v en u dn i>\ 2 _ fdn(u + v) cn(u + v)\ 2 

V snu sn v / \ sri(u + v) J 

by J. J. Thomson s formula of 121. 

142. But the Addition Theorem (1) for sn(u + v) of 116, 
sn u en v dn i> + sn v en u dn u 



1 
when translated into the inverse function notation, gives 



22? 
This reduces, for K = 0, to the trigonometrical formula 



the integral of 
and for /c = l, to 

tanh ~ l x + tanh ~ l y = tanh - l , 

l+xy 

the integral of cfo/( 1 - as 2 ) + c^/(l - y 2 ) = 0. 

Similarly, equations (2) and (3) of 116 may be written 



We can now see why so little progress was made with the 
Theory of Elliptic Functions, so long as the Elliptic Integrals 
alone were studied, and also why Abel s idea of the inversion 
of the integral has revolutionised the subject. 



G.E.F. 



146 THE ALGEBRAICAL FORM 

143. A slight change of notation in the canonical integral 
(11) of 38, suggested by Kronecker (Berlin Sitz., July, 1886), 
introduces a further simplification, on writing 

x = 
then dx/du = 



n /-t \ 

-j-9 = /C-( 1 -- )(1 KX) 

du 2 K 



with p = K - l -\- K - ) 

and now u=fdxj / JX, 

o 
with X = x( 1 px + x*). 

Now 

= sn - + sn 



144. In Weierstrass s notation, we take 

X = x*-g. 2 x-g^ 

so that, in the general expression of the quartic X, 
a = 0, 6 = 1, c = 0, d=-lg 2 , e=-g^ , 

and now Euler s form of the Addition Theorem becomes, with 
z for G the arbitrary constant, 



Now if x = $>u, y = $v, so that 
then we shall find ( 147) that 2 = >(u + 1>) ; so that 



or, in the inverse notation, 



Put i6+f = ty, so that 



since ( 51) #>w is an even function, and p w an odd function 
of w ; then, with 



therefore also, by symmetry, 



OF THE ADDITION THEOREM. 147 



Thus - 

$v $w <@w <@n> pu pv 

or ( pv - <$w }<>u + ( yw - $>u }<p v + (pu pv )$> w = 0, 
or (p v $> iv)pu + (p lv pu^pv + ($> u p v)pw = 0, 

1, pu, p u, 
or 1, pv, <@>v =0 ........................... (G) 



Weierstrass thus replaces the three elliptic functions sni<,, 
en u, dn u by a single function pu, and its derivative p u. 

145. Take for example the integral of ex. 8, p. 65, 
fX ~ $dx, where X=(x-a) (ax 2 + 2bx + c), 
a cubic function of x, having a factor x a. 
This example shows that we may put 

Z* ... ac-6 2 

^" Wlth ^=o, ^=^ 



a) 



and then 



Now, if y and are the values of x corresponding to the 
values v and w of u, and if 

= 0, or "Z - Sefo +Y~ kly +Z~ *dz = 0, 



then the integral relation (G) of 144 connecting x, y, z becomes 
(y-*)Z+(*-*)r4+( a! -y)Z*=0 ............ . ..... (1) 

We notice that the integral relation does not require the 
knowledge of the factor x a of X \ so that, writing 



we have, on rationalizing the relation (1), 



or X7Z={Aays+Btoz+zx+xy)+C(x+y+z)+I)}* ...(2) 

(MacMahon, Comptes Rendus, 1882 ; Q. J. If., XIX., p. 158.) 

Then 



sothat ff _ 



equivalent to Allegret s result (Comptes Rendus, 66). 



148 THE ALGEBRAICAL FORM 

146. We shall find it convenient to replace the constant O 
in Euler s integral relation by 4c + 4s, and to consider s as the 
arbitrary constant, the meaning of which is to be interpreted ; 
and then 



x-y 

or s = - * 

where 



F(x, y) = ax 2 y 2 + 2bxy(x + y) + c(x 2 + 4txy + y 2 
= (ax 2 + 2bx + c)y 2 + 2(bx 2 + 2cx + c% 
= (ay 2 + 2by + c)x 2 + 2(6^/ 2 + 2cy + d)x + cy 2 + 2dy + e, 
a symmetrical quadri-quadric function of x and y. 

Treating s as a function of the independent variables x and 
y t we shall find 

1 df ,y 1 dX , v 

~ ~ 



(ax* + 3bx 2 + 3cx + d)y + bx* + Sex 2 + 3cfcc + e 



9 /v , -c, /T7 

2 / Y_|_ M- -^.JJ suppose ; 

z * (x y) 
and similarly we shall find that \/Y has the same value. 
But if s is taken as constant, then 



or dxjJX + (%/</ F= 0, 

so that the differential relation which leads to Euler s integral 

relation is thus verified. 

147. But now denote 

4s 8 -02 s -0s b y 8 * 

where g% = ae 4<bd + 3c 2 , ^ 3 = ace + 26ccZ acZ 2 eb 2 c 3 , 
so that ( 75) g 2 and # 3 are the quadrivariant and cubicvariant 
of the quartic X (Burnside and Panton, Theory of Equations; 
Salmon, Higher Algebra). 



OF THE ADDITION THEOREM. 149 

We shall find, after considerable algebraical reduction, that 



(x-yf 
so that 1 ^4- l dy - l ds 

- 



and the elliptic elements dx/^/X and dy/*J Fare now reduced by 
this substitution to Weierstrass s canonical form ds/^/S of 50. 

Mr. R. Russell points out a concise way of performing this 
algebraical reduction, by means of the linear substitution 

t = (TX + y)/( T +l) in the quartic (a, b, c, d, e)(t, I) 4 ; 
which then becomes of the form 

Xr^4 i (X l y + X^ + QF(x t y^+4 ! (Y 1 x + F 2 ) T + F, 
or ^T 4 + 4T 3 + 6CT 2 + 4DT+#, suppose. 

If the invariants of this new quartic are denoted by (? 2 , G , 
then G, = (x - yYg G B = (x- y) 6 G B ; 

and $ = 4s 3 <s (r 



(x-y) 



(x-y)* 
148. Rationalizing the integral relation of 146, 



or s 2 (x - y) 2 - sF(x, y) - E(x, y] = 0, 

where E(z, y) = {(ac-b 2 ) 



+ i(ae - c 2 )2/ 2 + (be - cd)y + ce - d* ; 

<* ( -Artfc-yF-rffa y)-H(x, y) = o 

where ^(ic, y) = (ac b 2 )x-y 2 + (acZ bc)xy(x 



a symmetrical quadri-quadric function of x and y. 

149. When aj = y, F(x, x) = X, and 
^(aj, a;) = H(x t x) = (ac - b^ + 2(ad - bc)x s + (ae + 2bd - 

+ 2(be- 
the Hessian H of the quartic A r . 



150 THE ALGEBRAICAL FORM 

One value of s is now infinite, and the other 

t~ 

as in 75 ; for, when x = y, 

F(x,y)-JXJY = 

2(x-y) 2 

_ 1t {F(x,y)}*-XY . -ZE(x, y) H 

%x-yY{F(x, y)+JXJY}- ll F(x, y) + JXJY~ ~ X 
a substitution due originally to Hermite (Crette, LII., 1856). 
Now, since t = GO , when X = 0, or x = a, 



(x-y)* 

Q 



ft-\ -H/X), 

a denoting a root of the quartic X = ; and here 

T=*/(*P-9J-9J 

(I>+ YX-X+XY 

=lt _ 

(x-y)*{(Y } x+ 

where G is a certain rational integral function of x of the 
sixth degree, called the sextic covariant of the quartic X ; the 
preceding algebra showing that 

T 2 Z 3 =G 2 , or 4# 3 -# 2 //Z 2 +# 3 ^ 3 + 2 = 0, ......... (H) 

this is called a syzygy between X, H, and G. 

(Burnside and Panton, Theory of Equations, p. 346.) 
For instance, if X is already in Weierstrass s canonical form, 
so that, if x = $u, 

X = #/ 2 u = 4a; 3 - g 2 x - g s , 
then H= 

and now t = 

so that ^ = 



This may also be written 
pStt^ftt- 

150. Withy =00, 

2s = ax 2 + Zbx + c - 
or s 2 - (ax 2 + 2bx + c)s- (ac - 6 2 > 2 - (ad - bc)x - J (ae - c 2 ) = 0. 
With y = Q, 

2s = (ex 2 + 2dx + e- +Je^X)/x 2 , 
or cc 2 s 2 - (ex 2 + 2dx + e)s-(ae c 2 )x 2 - (be - cd)x -ce + d 2 = 0. 



OF THE ADDITION THEOREM. 151 

Writing F(x, y) in the first equation of 146 in the form 

Y+ J Y (x - y) + T V Y"(x - 2/) 2 , 

we can find x as a function of s and y by the solution of a 
quadratic, in the form 

..- 

This method of the reduction of the general elliptic element 
dx/^/X to Weierstrass s canonical form ds/^/S is taken from a 
tract " Problemata quoedam mechanica functionum ellipti- 
carum ope soluta. Dissertatio inauguralis" 1865, by G. G. A. 
Biermann, where the formulas are quoted as derived from 
Weierstrass s lectures. 

151. Changing the sign of ^/F, we find that 

.*fe.y)WA/r 

2(x-y)* 
leads to the differential relation 

1 dx 1 dy _ 1 ds f 
~~ = 



so 



9 

implying that u v = when x = y, since s oo when x = y : 
and now, in Weierstrass s notation, 



r 

that, putting /d( 

rx r*> 

u v = ldx\JX /ds/^/S, 
^/ +/ 



Changing the sign of v, and therefore again of F, 



so that p2it = - H X /X, $>2v = - H y / F, 

implying that u = when X = 0, v = when F=0 ; so that 



where a denotes a root of the equation X = 0. 
Then w 



152 THE ALGEBRAICAL FORM 

Mr. R. Russell finds, as is easily verified algebraically, that 



___ 
(x-yy X" (x-y)*X (x-yf Y" = 

But, from the Addition Theorem (F) of 144, 



and therefore 



2 p(uy)- 
the sign being determined by taking v small, when ^/ = a, nearly. 

Now, p / (u-i;) 



so that, as in 147, 



152. 

p2v = - It H y l Y = (6 2 - oc)/a, 
and p 2v = - It G v / Y% = (a?d - 3abc + 26 8 )/a l ; 

ax + b 



Again, from equations (F)* and (G) of 144, 

_ Y&+ Y 2 

~ 



and putting u = 0, and therefore x = a, we find 
aa -\- b _ p v + p 2v 
^/a <pv <p2v 

so that the quartic can be solved, when <@v and p v are known. 
(Solution of the Cubic and Quartic Equation, Proc. London 
Math. Soc., vol. XVIII., 1886.) 



OF THE ADDITION THEOREM. 153 



Otherwise, with t= -H/X, 

d^ = _H X-HX _ __2 
dx ~^~ X 

while T 3 = 4 3 - g z t -g s = G 2 /X 3 , 

so that d t/JT = - Zdx/JX, 

and 



u = /dx/^X = $/dt/JT= ^ ~ \ - H/X), 



a denoting a root of the quartic X=0. 

Then p2u = t = - H/X, & 2u = -T=- G/X? ; 
while v = when y = a, and Y= ; 

so that ptt = , = .^?) 

2(x a) 2 

u= /S= ( aaS + ^^ q2 + ^ Ca + ^ X + ^ 4 ^ Cq2 + IX 

(x-a) B ..** 

If v, k, K denote the values of u, s, S, when x = oc , 
^ = J (aa 2 + 26 a + c) = p v, J^ = (aa 3 + 36a 2 + 3ca + cZ)^/a = - p v 
7 aa 3 + 36 

S K = 

x a 
so that *_= __ = 



and now p2v = (b--ac)/a, p 2v = (a-d 

Conversely, given these values of p2v and p 2-y, and supposing 
the bisection of the argument of the elliptic functions to be 
carried out, we can determine %>v and p i , and thence solve the 
quartic equation X = 0. 

153. Since F(x, a) vanishes when x = a, a root of X = 0, it is 
divisible by x-a ; so that 



, suppose, 

x ~~ a 



a typical linear transformation, which converts dx/^/X into 
ds/^S, the canonical form of Weierstrass. 

Denoting the four roots of X = by a, /3, y, S, then since 
6/a= -i( a 
we may write 



= jL f a- 

x-a a -/3 a-7 a-6V 



154 THE ALGEBRAICAL FORM 

and now 

= /3(a - y)(a - 8) + y(a - 3)(a - ) + ffa - /3)(a - y) 



a 
with three other values ft, y , S corresponding to ft y, S. 

Now ^/S=- -j-^- TO 

(x-aY 

JX 



^^ 

Denoting by e 1} e 2 , e 3 , the roots of the discriminating cubic 

4<e*-g 2 e-g 3 = Q, 

so that $=4(s gjXs e 2 )(se 3 ), 

then we may write 

r _ Q 

s- ei = \a(a -y)(a-S ) ^, 

C OL 



JU OL 

so that, to x = a, /3, y, S, corresponds s GO, e v e 2 , e 3 ; and then 

- y) }, 



If we interchange a and /3, and put 



then to z = /3, y, (5, a, corresponds s 1 = oo , 6 3 , e 2 , ^ ; 
so that s = s l gives a linear substitution converting 

dx/^JX into dzjJZ, 
in which 05 = a, /3, y, S, corresponds to z = /3, y, ^, a. 

If s is replaced by pu, and the same function of z by p;, then 
we find from 54 that 

V = U, U + 0) v U + CO-L + ft) 3 , U + 2ft)! + fc> 3 , 

gives the four linear transformations which leave dx/^/X 
unaltered ; and corresponding to the values (a, /3, y, 8) of x 
we find (a, ft y, 5), (A y, 5, a), (y, A a, /3), (ft a, ft y) of ; 
the first transformation being merely z = x, not a distinct trans 
formation. 



OF THE ADDITION THEOREM. 155 

154. When, as at first, 



_ 



and when e is a root of the discriminating cubic, then s e is a 
perfect square ; and we find 



where, as in 70, the quartic X is resolved into the quadratic 
factors N x and D X) and Y into the corresponding factors N y 
and D y ; this can be done in three ways, corresponding to the 
three roots of the discriminating cubic. 
Thus the integral relation 



x-y 
leads to the differential relation 

as is easily verified algebraically, N and D being quadratics. 

155. A more elegant expression can be given to these rela 
tions if we follow Klein (Math. Ann., XIV., p. 112 ; Klein and 
Fricke, Elliptische Modulfunctionen, 1890) in employing 
homogeneous variables x and x. 2 , by writing xjx 2 for x, and 

2/1/2/2 f r y > an d now 

/dx 
7x~j 



Conversely, by writing x for x v and 1 for x 2 , we return to 
our original non-homogeneous variable x. 

Klein employs the abbreviations 

(xdx) for x z dx l x 1 dx z , and (xy) for x^ 
also fx for (a, 6, c, d, e)(x v # 2 ) 4 ; and now with 



156 



THE ALGEBRAICAL FORM 



and 



reducing to the above in 153, when fz/ = 0. 

The Hessian JT or ZT( 15 03 2 ) of X or f(a) 15 a; 2 ) is now given by 



and the sextic covariant G or G^, a5 2 ) by 



aff 



We may also use x and y as the homogeneous variables in 
the quantities, instead of x and x 2 . 

Thus, for example, the integral ff~&(xdy), where 

f = x ll y -f 1 lx Q y Q xy 11 (the icosahedron form) 
is shown to be elliptic by means of the substitution 



where 



dxdy 
Then we can verify the syzygy 

where T= - 



Now 



since 



g s =^T 2 ~ 5 , provided <? 3 = 



OF THE ADDITION THEOREM. 157 

f l xcly 6 f x dz 6 6 

j w=u^^^=^- iz -^ m ^ 

Similar reductions will show that the interals 



are also elliptic ; also the integrals 

f(tfy - xy 5 ) ~ \xdy) and 
depending on the octahedron form, 
(Schwarz, Werke, II., p. 252 ; Klein, Lectures on the Icosahedron.) 

"- ! 

156. The further development introduces the theorems of 
Higher Algebra on the quartic and cubic, for the treatment of 
which the reader is referred to Salmon s Higher Algebra and 
Burnside and Panton s Theory of Equations. 

Thus, H denoting the Hessian of a quartic X, and e v e 2 , e 3 
the roots of the discriminating cubic 

4e 3 g 2 e g B = 0, 

then 4(H+e l X)(H+e,XXH+e.,X) = 4 ! H 3 -g,HX- 2 +g B X s = - G\ 
where G denotes the sextic co variant ( 149) ; so that H + eX 
is the square of a quadratic factor of G. 

Following Burnside and Panton (p. 345) we shall find it 
convenient to put 16(H-{-eX)= P 2 ; and then 

P 1 P 2 P 3 =32G, 
P p P. 7 , P 3 denoting the quadratic factors of the sextic co variant G. 

Then P, 2 + P 2 2 + P 3 2 = - SH, 

since e x -f e 2 + <? 3 = ; 

while (e, - e 3 )P^ + (e, - eJP,* + (e, - e 2 )P 3 2 = ; 

and eJP* + e 2 P 2 2 + e 3 P 3 2 = - 1 6(^ 2 + e* + ef)X = - 8g. 2 X. 

Since (e 2 - e 3 )P 1 2 = (^ - e 3 )P 2 2 - (^ - e 2 )P 3 2 



therefore each of these factors must be the square of a linear 
factor, and we may therefore put 



so that Uj and u 2 are linear ; and now 



158 



THE ALGEBRAICAL FORM 



157. Mr. R. Russell points out (Q. /. M. t XX, p. 183) that 
Hermite s substitution of t = H/X reduces the integral 

dt 



cu &ur , 3 _u 

dx~ "Jf 2 an ~~^ 2 ft 2? 



For 

so that G~^dx = J(4 3 g z t g 3 }~%dt. 

Again the integraiy*(4 3 g 2 t g B )~%dt, as well as the general 

(2) 

.(3) 
where K or K(x, y) denotes the Hessian of the cubic U(x, y), 



integral 

where 7 or U(x, 1) denotes the cubic (a, b, c, d](x, I) 3 , 

is again proved to be elliptic by the substitution 



given by 



, y} = 



dxdy 



"dotty* 



The cubicovariant J of the cubic U is given by 



,(5) 



da? 1 dy 

and the discriminant A by 

A = a 2 ^ 2 + 4ac 3 -6a6ccZ+4c^ 3 -36 2 c 2 ; (6) 

and now we have the syzygy 

(Salmon, Higher Algebra, 192; Burnside and Panton, 
Theory of Equations, 159.) 

Differentiating (3) logarithmically 

3ds_3JT_2T_ _&7 
sdx~ K " U ~~ KU 
J 



while 

so that 
and 



dx 
U* 



sdx Uds 

-K = ~T 



ds 



(8) 



OF THE ADDITION THEOREM 



159 



When we know a factor, x a, of V, then we may employ, 
as in ex. 8, p. 65, the substitution 

s=U*/(x-a) ............................. (9) 

Putting U= (x - a)(ax 2 + 2V x + c ) 



then 4;2 3 g s is a perfect square, when 

ac -V 2 



93 ~ 



and now 



z = 



00 s + 26 a + c ~ aa 2 + 26a + c 
aa; 2 + 26 ar + c - # 3 (x - a) 2 







-3K 



3s 



(aa 2 + 26a -f c) U% ad 2 + 26a + c 



while 



9/ 



9(4s 3 + A) 



(aa 2 + 2ba + c) 3 ?7 2 ~ (aa 2 

, ............. (10) 



a transformation equivalent to that of 47. 

158. Mr. R. Russell also shows (Proc. L. M. S., XVIIL, p. 57), 
,, 



where X denotes a quartic and H its Hessian, can be reduced 
to the sum of three elliptic integrals by Hermite s substitution 

t=-H/X. 
For we may replace ( 156) 

lx z + 2mx + n by 2xF\ + gP 2 + rP 3 

or by 4p V( -H-ej:) + 4gV( ~ ^- ^0+ 4 ^ ( -JET- e 8 Z), 
where ^9, g, r are determined by equating coefficients ; while 



so that the integral becomes 
fp^-H-e^+q^-H-e^+rJt-H 
J aX + 3H . a X + H 



JXdt 
/3H 



160 THE ALGEBRAICAL FORM 



the sum of three elliptic integrals. 

Particular cases may be constructed by making /3 and /3 
zero, or a and a zero ; when we obtain 

f(lx 2 + 2mx + n)dx/X, or f(lx 2 + 2mx + n)dx/H. 

159. Mr. Russell remarks that the reduction of the well- 
known hyperelliptic integral 

r (Ix^ + ^mx+ri^dx 

J x /( 1 X 2 . 1 + KX 2 . 1 + \X* . 1 K\X 2 ) 

to the sum of elliptic integrals is a particular case of this 
theorem, since the quartics 

1 x 2 . 1 K\x 2 and 1 + KX 2 . 1 + \x 2 

can be expressed in the forms aX + (3H and a X + fi H, 
by taking X= l+/cAa? 4 , and therefore H= K \x 2 ; 
and now a = l, a =l, fi= -(l+/cA)//cA, /3 X = (/c + A)//cA. 

These integrals are considered in Cayley s Elliptic Functions, 
chap. XVI., where x 2 is replaced by x; they arise in the expres 
sion of Legendre s elliptic integral 

jd<pl&((f), b) in the form E-\-iF, 
when the modulus b is complex, so that b 2 = e+if. 
(Jacobi, Werke, I., p. 380 ; Pringsheim, Math. Ann., IX., p. 475.) 
Writing P for x(I-x)(I+ K x)(l+\x)(l- K \x), Jacobi finds 

>, c)+F(<i>, 6)}, 



where 



or 



x 



2( 5 

A W ft - 



OF THE ADDITION THEOREM. 



Then employing the inverse function notation, 
fdx 
J P 



I ( ll+K.l+\.Xj\ . ll+K.l+X.X 

" 



/xdx _ 
?~ 





When X is negative, then b and c are conjugate imaginaries ; 
so that we can now express F((/>, 6) in the form E+iF, when 
6 2 is of the form e + if. 

For, writing - X for X, and now writing 

P for x(l-x)(l+ K x)(l-\x}(l -\-K\X\ 



f 2E f xdx - 

J JP *J(1+K.1-\) J JP 



th 

In the particular case considered by Legendre, X = 1, and now 

P = x(l-x^(I-A 2 ), 
on replacing K by /c 2 ; so that 



-x 2 . 1 - 

can be expressed by elliptic integrals. 
Mr. R. Russell employs the substitution 



and now 

dy f A(l-Bx)dx 



so that, putting 

x{(I+Bxy 2 -Ax}{(l+BxY--o-Ax}=P, 
therefore & = K -\ 2 , B= J( K \). 

Taking J B = x /(/cX), and 

) 2 - Ax = (l-x)(l- K \x\ 



then 2J K \-A=-1- K \, 



and taking B= ^/(/cX), 

G.E.F. L 



162 THE ALGEBRAICAL FORM 

160. Mr. Roberts s integrals (Tract on the Addition of the 
Elliptic and Hyperelliptic Integrals, p. 53) 



where Q is a reciprocal quartic in x 2 , say 

or aQ = (ax* + %bx 2 + a) 2 - (2a 2 + W - 

furnish another particular case of Mr. Russell s theorem, since 
Q can be expressed in the form 



where X and H are in their canonical forms, 



y = dv 



Or we may put x-\-x l = u, x x~ l = v, when the integral 
becomes $A (U+V) + B( U- V\ 

where U= f du 



Thus 



where X = I+x\ H=x\ 

Therefore the 



integral / 



j -- ^- 

is reduced to elliptic integrals by a substitution, such as 

y = (I+x^/x 2 ; 
and then becomes 



Another particular case of the general theorem occurs in the 
reduction of the integral 



where R is a sextic function, the roots of which form an involu 
tion, and whose invariant E therefore vanishes (Salmon, Higher 
Algebra, 1866, p. 210). 



OF THE ADDITION THEOREM. 1G3 

This invariant E is the one tabulated in the Appendix, 
p. 253, Higher Algebra, where it occupies thirteen pages. 

The sextic covariant G of a quartic X is a specimen of a 
sextic of which the roots form an involution ; and writing 
32 G or 



c 2 ) (a 3 # 2 + 2b s x + 1 3 ) 

= a 1 .>. l .x 1 a 2 C "~ ^2 x ~~ ^-2) a s( x O 3 .x 3 ), 
then since the squares of P p P 2 , P 3 are linearly connected by 
the relation of 156, therefore P v P 9 , P 8 are mutually har 
monic, and any one is therefore the Jacobiari of the remaining 
two ; this leads to the three relations 

a. 2 c 3 -f 3 c 2 26 2 6 3 = 8 c 1 + c^Cg 26 3 6 1 = c^c., 4- a. 2 c t S/^i^., = 0. 



^ 

7 = 



are the six linear transformations which reduce 

to 



as in 74 ; so that if the quartic A" is resolved into the 
quadratic factors JV and D, we may write 



Now N/D is maximum or minimum when x = 9, or 0. 

Making P 1? P 2 , P 3 homogeneous by the introduction of y, 
which is afterwards replaced by unity, so that 

P=(a v b 1 ,c l )(x, y y- ..... 

then the three distinct linear transformations of 153, which 
leave dx/^/X unaltered, are found to be 



^ _ _ 

KB J " ltyVx ~ty ~dx 
(R, Russell, Proc. L. M. S. t XVIII, p. 48.) 

Now i dx or f( Au i+ Su J( u 2 du i - *VK) 

y *JG J ^/{^^(V-V)} 

where u 1} u. 2 are defined in 155, is reduced by the substitution 
y 2 = uju v or p(x-<f>\ (x-0), 



164 THE ALGEBRAICAL FORM 

This integral has been considered by Richelot (Crelle, 
XXXIL, p. 213) ; and by differentiation we find 



according as ?/ 2 is less or greater than ^/2-l ; and thence the 
integration can be inferred; the value of K to be taken is 
- 1 or tan 22 J, when it will be^found that K jK = 

161. As further applications, consider the integrals 



where A0 = ^(1 - 6 2 sin 2 </>). 

(Legendre, Fonctions elliptiques, I., p. 178.) 
Putting A0 = ; 2 , and l-Z> 2 = c 2 , then 



1 

the integration required in the rectification of the Cassinian 
oval, given by 



or 



where r lt r 2 are the distances from the foci (a, 0). 

The expression 1 x*.x^ c 2 can be expressed by H 2 
where X = x* + c, H=(l+c)x 2 ; 

and now the substitution y = X/H gives 



so that 

-16 



V(2+2 C ) 

by means of the results of 39-41. 
In the Cassinian 



OF THE ADDITION THEOREM. 165 

rZs 2 2 r 2 



8 



Now, if we put 

r* = (a 2 +/3 2 ) 2 cos 2 + (a 2 - /3 2 ) 2 sin 2 0, 
then s = a 2 /{ a 2 + /S^cos 2 ^ + (a 2 - /3 2 ) 2 sin 2 } 



Similarly 



which can be expressed in a similar manner. 
Again, substituting A 2 = # 3 , then 





particular cases of the preceding general integrals. 

Mr. R. A. Roberts (Proc. L. M. S. t XXII., p. 33) has shown 
that /(lx + m)(ax 6 + 2btf + c) ~ s or - j^ 

can be expressed as the sum of elliptic integrals, not always 
however in a real form. 

Mr. Russell shows that if x O v x 6. 2 are the factors of P v 
a quadratic factor of the sextic covariant, then 

lx+m 



is reduced by the substitution 

l f= p ( x -o i )/ (x ^e. 2 ) 

to the form lr-. R ^ * . , . dy, 

*//(a2/ 8 +26?/ 4 +c) " 

and this again by the substitution 

to the forn, 

two elliptic integrals, not necessarily however in a real form. 



166 THE ALGEBRAICAL FORM 

Abel s Theorem, applied to the Addition Equation. 

162. Euler s Addition Theorem is now found to be a very 
special case of a Theorem of great generality, due to Abel, the 
method of which we shall employ here, in the very limited form 
required for the Addition of the First Elliptic Integrals. 

Consider the points of intersection of the fixed quartic curve 
whose equation is 

2/ 2 = A , ................................ (1) 

with any arbitrary algebraical curve whose equation in a 
rational form may be written 

f(,2/) = ................................ (2) 

By continually writing X for y* 2 , we can reduce equation 
(2) to the form P + Qy = 0; ............................ (3) 

and now the abscissas of the points of intersection of (1) and 
(2) are given by the equation 

P + QJX = 0, ............................. (4) 

or, in a rational form, P 2 Q 2 X = .............................. (5) 

Denoting the degree of this equation (5) by /*, and its roots 
by x v x 2 , ... Xfi, Abel puts 

^x = P 2 -Q 2 X = C(x-xJ(x-x 2 )...(x-Xn), ......... (6) 

and now he supposes the roots of this equation to vary in 
consequence of arbitrary variations in the coefficients of the 
terms in equation (2), corresponding to arbitrary changes in 
the shape and position of this curve ; the coefficients in 
equation (1) are however kept unchanged. 

If dP, 3Q denote small changes in P and Q due to the 
changes in the coefficients, and if dx r denotes the correspond 
ing change in any root x r of equation (5), then 

Vx r . dx r + 2P<5P - 2Q8QX r = 0, 
or, making use of equation (4), 

= 0, 



dx r _ 9 QdP-P8Q_Ox L 

- ~ - 



suppose. 

Now, if the degrees of P and Q are denoted by p and q> 
then the degree of Ox is p + q , and we shall find this is 
always at least one less than JUL I, the degree of i//#, or two 
less than /m, the degree of \fsx. 



OF THE ADDITION THEOREM. 167 

For if in equation (3), P 2 and Q 2 X are of equal degree, then 
q=p 2, and n = 2p , so that /x p q = 2; and pp q is 
greater than 2, if q is less than p 2. 

But if q is greater than p 2, then the order of \/ric is given 
by that of Q 2 X, and therefore /uL = 2q + 4<, while p = + l at 
most ; so that /u. p q = 3 at least. 

Since x6x is thus of lower degree than \fsx, we can split the 
fraction x6x/\fsx into a series of partial fractions, such that 

r x r 9x r m 



and now, if we make x 0, we find that 



a theorem in Algebra due to Euler ; otherwise stated as 

/y. W 

_ = 0- ..... (9) 



provided m is less than /z 1, the * marking the position of 
the missing factor x r x f . 

Applying this theorem to equation (7), we find 



so that, if, in consequence of any finite alteration of the 
coefficients in equation (2) or (3), the roots of equation (5) 
become changed to x\, x 2 , ..., x ^, then 

aX + ... +^dx^ = 0, ...... (11) 



the Theorem of Abel, as required for present purposes. 

It is the combination of the theory of Integrals and of the 
theory of Algebra which furnishes the key of Abel s Theorem ; 
the algebraical laws are expressed very concisely by a single 
equation (5), of which the variables are the roots, and whose 
coefficients are not independent, but are connected by a number 
of relations. 

Thus, if we take P of the ^ th order, and Q of the order p- 2, 
we have a plexus of ^ or 2p equations of the form (4) 



and the elimination of a, /3, /,..., y, ... leads to a determinant 
of 2p rows, each row of the form 



h--2 r /t- 

Jb r , . . . , JU r 



168 THE ALGEBRAICAL FORM 

163. Suppose for instance that (2) is the parabola 



(2) or (3) 
then equation (4) becomes 

ax z + 2px + y-JX = 0, ...................... (4) 

and (5) becomes the quartic equation 

(ax*+2px + 7 ) 2 -X = 0, ...................... (5) 

Denoting the roots by x v x 2 , x 3 , x^ then the elimination of 
a, /3, y leads to the determinant 



4 > *^4 *i X/ 4 

as the integral relation, corresponding to (/JL 4), 

& 



By making a = ^/a, so that the parabolas are of constant 
size, or by writing equation (5) in the form 



one root, x suppose, becomes infinite ; and now 

4(/3 - % 3 + (4/3 2 + 2ay 
so that 

= 6c - 2y - 



or 

Now the two relations 

^ + y - Vx/- Y i = 0, 
+ y - JaJX t = 0, 
give by subtraction 

(x, - x z ){a(x 1 + x 2 ) + 2/3} = V 

S 



where (7 = 2aa; 3 2 + 46^ 3 + 6c 2 x /a /v /A r 3 ; 

and we thus obtain Euler s original integral relation, the 

general integral of the differential relation 

dx 1 /^/X l + dxJ^/X 2 = 0, 

when is constant ; and a particular integral of 
c^/V^i + dx 2 /JX 2 + dxJJXs - 0, 
when ic s is considered as variable. 



OF THE ADDITION THEOREM. 169 

164. When X is in Legendre s canonical form 1 x 1 . l 
then Abel takes P = ax + x*, Q = b , 

and now equation (6) becomes 

- K 2 x-) 



where xf + x. 2 2 + x/ = b 2 K 2 - 2a, 

xfxf + x/x^ + a;^ 2 = b 2 + 6V + a 2 , 

r 2 r 2 r 2 _ 7,2 
l X -2 X 3 - U . 

But a and b are determined by the equations 

ax l + ^j 3 + 6Z X = 0, aa3 2 + xf + 6.Y 2 = : 

so that b 



and therefore, as in formula (1), 116, 



> 
^ 3 a^! JT , x. 2 X l 1 K-x^xf 

Also 1 - r/Y 2 . 1 - x, 2 " 1 - 3 2 - 1-6 V + 2tt + 6 2 + b 2 K - + a 2 - 



while ^ j 2 + a/V 2 + # 3 2 K*xfafa = 2a, 
so that 

2 - a^ 2 - o; 2 2 - a; 3 2 + &x*xx = 2(1+ a) 



or (2 - x* - x* - xf + K*-x*-xx 3 *y = 4(1 - .r^Xl - 2 2 )(1 - ^ 2 ), 
which may also be written 



as in 119, with ^ 1 = snit, x. 2 = snv, x 3 = su(uv). 

This, with x^=siiu l , x 9 = snu 9t o: 3 = sn^ 3 , may be written 
1 cn 2 i^ cn 2 i6 2 cn 2 ii s + *2 en i^ 
where u^ + it 2 

( 131); and, with a triangle of Class I., is equivalent to the 
formulas in Spherical Trigonometry 

1 cos 2 a cos 2 6 cos 2 c + 2 cos a cos b cos c = /c 2 sin 2 a sin 2 6 sin 2 c 
= sin 2 ^. sin 2 6 sin 2 c = sin 2 a sin 2 jB sin 2 c = sin 2 a sin 2 6 sin 2 (7. 

165. To obtain the Addition Theorem for Weierstrass s 
functions, we consider the intersections of the cubic curve 

y* = x*-g&-g s , or X, ............... (1) 

with an arbitrary straight line 

(2) 



170 THE ALGEBRAICAL FORM 

Now, if x lt x 2 , x% denote the roots of the equation 

then 

so that a = v ^_ v ^ 2 , ft = ^ 

and ( 144) a; 1 +aj 2 +c 3 = 

The elimination of a and ft between these two equations and 

leads, as in 144, to the determinant (G) 

1, x lt \/X l 1, pu, $ u 

1, x 2 , ^/X% =0, or 1, pv, <@ v 

1, x , ,JX 1, 

where u + v + iv = 0. 

In addition, from (5), 

so that 

166. Consider the intersections of the fixed cubic curve 
with a variable straight line 

Then \}sx = (ax + /3) 3 ( Ax* + 3Bx 2 + 3Cx + D) 



and 



- - 3 



/Y> ,7 /y> _ r^ 

a 3 -^ 
Denoting by y v y z , y. 3 the corresponding values of y, then 



{ C+ J (a 8 - 



j + x 2 + a; 



as in 145. 



OF THE ADDITION THEOREM. 171 

Now, if the constants a and /3 receive small increments 
Sa and S/3, then 

yf/x^dxi + 3(00?! + /3) 2 ( V + Sfl) = 0, 
and \/s x l = (a 3 A )(x l X 2 )(x 1 x s ), 

dx, 




=0, 

and the sum of the three integrals is a constant, which can be 
made to vanish by taking for the lower limits a root of the 
equation y = 0. 

In the particular case of the cubic curve 



the relation expressing the collinearity of the three points is 

aj 1 a? 2 ar s + 2/i2/ 2 2/3 =1 - 
Now, as in 145, with # 2 = 0, # 3 =1, and 



and, by symmetry, with 

(i 



we find from (F) 144, after reduction, 



so that it + ! = , a constant. 

With pa = l, then ( 149) p2a = l ; so that ( 62) 

p2a = ?(2a> 2 ~ a )> or a = ^2- 
We may therefore put 

U = Jct> 2 + , V = Jft) 2 #, 

and express x and ^/ by functions of t. 

For any other arbitrary value of a, the integral relation 
connecting x and y will be, by 145, 



and treatin as constant, this leads to the differential relation 

= 0. 



172 THE ALGEBRAICAL FORM 

We can put 

(1-g*)* _(l 
-~ p * ~ 



where 

and pi0 = l, for the value z= oo ; and then 



167. When the quartic JT is resolved into two quadratic 
factors N and D, we may replace (1) by the quartic curve 

y*=N/D; ................... ........... (i) 

and now equation (4) is replaced by 

PJD + QJN=0; ......................... (4) 

so that equation (5) becomes 

P*D-Q Z N=0 ........................... (5) 

The elimination of the constants from the plexus of equations 
determined by the roots of this last equation (4) leads to 
determinants, whose rows are of the form 



For instance, by taking P and Q linear, so that the variable 
curve (2) or (3) in 162 is a hyperbola, we can obtain the 
integral relation of 154 in the form 



- eontant . 

(W. Burnside, Messenger of Mathematics.) 
We have taken X as a quartic function of x, so as to apply 
to the elliptic functions, but Abel s theorem holds for any 
higher degree of X, the method of proof being exactly the 
same; and, according to Klein, we resolve X, supposed of 
even degree, into factors N and D, differing in degree by or 
a multiple of 4, when we wish to make use of the fixed curve 



168. The reader is referred to the treatises of Salmon or of 
Burnside and Panton for the proof of the Theorems in Higher 
Algebra quoted here ; they are easily verified, however, if we 
work with the quartic in its canonical form 

U= x* 6m x 2 y 2 + y 4 ; 
when H = mx* + (1 3m 2 )x 2 y 2 my*, 

G = J(l - 9m 2 )xy(x* - y*). 



OF THE ADDITION THEOREM. 

The following examples, taken from recent examination 
papers, will illustrate the character of the algebraical work. 

EXAMPLES. 

1. Denoting by U the binary quartic, reduced to its canonical 
form, # 4 6ra# 2 i/ 2 -f y 4 , its quadrin variant and cubinvariant by g. 2 
and (/ 3 , and its Hessian and sextic covariant by H and G, 
prove that 

(i.) 4m 3 -# 2 m-(7 3 = 0; 
(ii.) H-t-mU is a perfect square ; 
(iii.) 



O Tf 

(iv.) H, ~ 



(vii.) the Hessian of XU+imH is 

(X 2 - T V(7X)ff + (i 

and the sextic covariant is 



2. Denoting the roots of 4e 3 g 2 eg 2 = Q by e v e. 2 , e s , prove 
that the roots of (x- + %g. 2 } 2 2g 3 x = 
are of the form 



3. Denoting the discriminant, Hessian, and cubico variant of 
a cubic / by A, K, and /, prove that 



(Work with the canonical form U=ax 3 -\-by 3 .) 
Denoting the same functions of XJJ-h/xG by A , K , J , prove 
that A = (X 2 -/A 2 A) 2 A, 



4. Prove that X and Y in 139 have the same invariants 
and g s (Burnside and Panton, 1886, p. 418). 

5. Prove that, in 156, 

Vfe - 8 >Pl + Vfe - *l) 

is the square of a linear factor of X. 



174 ALGEBRAICAL FORM OF ADDITION THEOREM. 

6. Discuss the properties of the quartic X in 153, whose 
roots are a , ft , y , 3 . 

7. Prove that ( 160) O lt </>! ; 2 , 3 ; $ 3 , 2 > define an involu 
tion of the roots of the sextic covariant G (R. Russell). 

8. Prove that the cubic substitution 
y = (bx s + 3e$ 2 + 3fZo; + e)/(ax s 

dy 3dx 

makes j-, ^7-^ TT~\ = 

^ (fjMyg^Uy) 

where U x =(a, b, c, d, e)(x, I) 3 . 

(Hermite ; Crelle, LX., p. 304 ; R. Russell, Proc. L. M. S. t 
XVIIL, p. 52.) 

dx 



9. Integrate /^4r 

10. Prove that, with s = pu, 



u - e) - - (s 2 - 268 - 2e 2 
- e 



2 U = 8 -f 1 ~ ^ 1 *L_- ? 



s e 2 

11. Prove that, if 

(i.) p(v; -20, -40) = 5, then p2v = 0, ^3v= 
(ii.) p(v; -60, -10) = 5, ............ 0, ......... i, 

(iii.) KO; -15, 19) =f, ............ I, ......... W, 

12. Prove that 

(i.) /(A + Bx)dx/y is elliptic, if y 2 = (1 - x 2 )(a + 3a? 

(ii.) f(A+Bx+Cy)dx/lj- is elliptic, if 

t/ 
f(a;, y) = (a, 6, c, /, gr, ^)(^ 2 , 2/ 2 , 1). 

(W. Burnside). 



CHAPTER VI. 

THE ELLIPTIC INTEGRALS OF THE SECOND AND 
THIRD KIND. 

169. The Elliptic Integrals, and thence the Elliptic Func 
tions, derive their name Elliptic from the early attempts of 
mathematicians at the rectification of the Ellipse. 

It was some time before mathematicians perceived that the 
simple integral to begin considering is 



which has not originally such a special connexion with the 
ellipse ; but the name Elliptic Integral has nevertheless been 
retained generally for all integrals of this nature. 

To a certain extent this is a disadvantage ; not only because 
we employ the name hyperbolic function to denote coshi(, 
sinl^t, tanh u, ..., by analogy with which the elliptic functions 
would be merely the circular functions cos<, sin <p, tan 0, ...; 
but also because it is found that the elliptic functions are a 
particular case of a large class, called hyperelliptic functions, 
but included in a larger class, called Abelian functions after 
Abel, which, beginning with the algebraical, circular, hyper 
bolic, and elliptic functions of a single argument u (jj = l) 
are in the general case the functions of p arguments which are 
met with when we consider the integrals 

/(!,, a* ..... x"- 1 ) dx/JX, 

arising in the linear transformations of J"dxj^/X, in which 
X is a rational integral function of x of the degree 2p + 2: 
for now the linear transformation (lx+m)/(l x+m f ) converts 

into (lm -V 

175 



176 THE ELLIPTIC INTEGRALS 

170. Legendre s elliptic integral of the second kind has already 
been defined in 77 ; and denoting it by E$, then the length 
of the arc BP of an ellipse is given by aEtf>, where the arc BP 
and the excentric angle of the point P are both measured from 
the minor axes OB, and now the modulus is the excentricity of 
the ellipse. 

The quadrant of the ellipse BA is given by aE, where, 

r^Tr 
as in 77, E denotes /A0c?0, the complete elliptic integral of 

o 
the second kind, in which < = JTJ-. 

The perimeter of the ellipse is therefore ^aE, the same as 
that of a circle of radius aEfyir. 

The periodicity of sin < and A< shows that, as in 14, 



and generally 

when m is an integer. 

Expanded in ascending powers of the modulus 



2 - 

so that, employing Wallis s theorems of integration, as in 11, 

T^ fj* i fi n ^?/1.3.5...27i-lV K- H H - 

E=J^ = ^\l-^^-^~^) -_J, 

o n ~ l 

whence the numerical value of E can be calculated. 

Tables of the numerical values of E(j> for every degree of < 
and of the modular angle are given in Legendre s F.E., II., 
Table IX. ; while the values of log E are given in his Table I. 
for every tenth of a degree in the modular angle. 

We reproduce this Table of logE, and of log^ 7 , correspond 
ing to the complementary modulus AC , to 7 decimals, and to 
every half degree in the modular angle J, corresponding to 
the values of logK in Table I, p. 10. 

171. By differentiation and integration, we prove that 
d(E<j>\ F<j> d f .. fd<t> E<j> K 2 sin cos 0. 

zk?) ? d-^ F * }= J iv"^-?*- AT 

and therefore, with $ = JTT, 

dE K d E 



OF THE SECOND AND THIRD KIND. 



177 



o p o p >p p o p p 9 p p 




o Lt o >p p ip p ip o o p o p o o o o o p ip p ip o o p o p o 9 H 9 



! ip 9 o 9 ip 9 p 9 p 9 p 9 ip 9 ip 9 p 9 p 9 ip 9 ip 9 ip o p o p 9 >p 

1 4n Th cc ci A, ^^-iHobbcixoc^t^iioui^^rcrc^c^^bbb 

N i>i>-i>i>-i>- i>-i>t t-i>.occooo^oooco;:;ooocooooccoo 



w 

H^ 



f illlliill||lilll|liis|g|gfii; 



i cot^ ~ o~ ^T^ rc?Ji p-Ho ~ < SSt>^S>O < ^cOTO<M-^oSSt^cD 

Icoco co QCGO oooo ccoo r^r^r^r^r^t^r^r^t--r^ 



p 9 p 9 o 9 p 9 o 9 ip 9 >p 9 ip 9 p 9 p 9 p 



9 p 9 o 9 p 9 p 9 o 9 ip 9 ip 9 p 9 p 9 o 9 ip 9 p 9 !p 9 ip 9 ip 9 

O CT5 CS CC 3C t^* r^* tO sO O O "^ "^ CC CC C^l ^ ^^ ^ O O C^ C^ CO CO t>* t^* SO CO 1^5 Li 

Ciooa5ooxxxxQccoxcoooooGocoxoooooooor^i^t^i> r^t^t> j> t^t^ 



o o o b o 
o o o o o 






099909999999 



tc 



G.E.F. 



178 THE ELLIPTIC INTEGRALS 

We can now prove Legendre s relation, that 

EK +E K-KK is constant, and = JTT ; 
for denoting it by A, we find that dA/dic = Q, so that A is 
independent of K ; and taking K = 0, then 



o o 
172. In Jacobi s notation, with < = 

E(f> = E am u =fdu. 2 udu ; 

o 
and now, from the quasi-periodicity of am u ( 14), 



where m is an integer. 

We may therefore, as in 78. separate E&mu into two 
parts, one the secular part, increasing uniformly with u, at a 
rate 2E per increase 2K of u, and the other a periodic part, 
denoted by Zi& in Jacobi s notation, and called the Zeta 
function ; so that 



or Zu =/(dn% - EjK)du. 



Addition Theorem for the Second Elliptic Integral. 

173. A well-known theorem, due to Graves and Chasles, 
asserts that if an endless thread, placed round a fixed ellipse, is 
kept stretched by a pencil, the pencil will trace out a confocal 
ellipse (fig. 22). (Salmon, Conic Sections, 399.) 

If the excentric angles (measured from the minor axis of the 
ellipse) of the points of contact P, Q of the straight parts of 
the thread PR, RQ are denoted by <j>, \[s, so that the 

arc BP = aE(f), arc BQ = aE\js ; 

and if we put <p = am u, \}s = am v, the modulus K being the 
excentricity of the ellipse, then, as asserted in ex. 6, at the end 
of Chap. IV., R moves on a confocal ellipse, when u v is 
constant, and conversely. 

For the coordinates of R being given by 

cos \!s cos (h 7 sind> sinx^r 
x = a r-h- -- r-rS y = b .^ r\ 
sm(0 \Is) sm(0 \/s) 

we find from Jacobi s formulas (4), (5), and (31), 137, replacing 
u and v by ^(u + v) and ^(u v), 



OF THE SECOND AND THIRD KIND. 



cni> i 






sin (am u ami;) 
sn u sn v 



__ _ 

" " 



sn 



179 
-t;) 



am-?;) 




Therefore 



Fig. 22. 



, cn 



a = a dc ^ ~ 



where 

so that a 2 -/5 2 = a 2 -6 2 , 

and therefore E describes a confocal ellipse, if u v is constant. 

If 16 + v is constant, 

-we find (x/aj-(y//3r = l, 

where </ = a K sn |(u + v), $ = a K en l(u + v), 

so that a /2 + /3 2 = a V - a 2 - 6^ 

and ^ therefore describes a confocal hyperbola (MacCullagh). 

To realise mechanically this motion of R on the hyperbola, 
the threads RP, RQ must pass round the ellipse, and be led, 
in the same direction, round a reel rnoveable about a fixed 
axis at C ; so that, as the reel revolves, equal lengths of thread 
are wound up or unwound. 

If the hyperbola starts from the ellipse at L, then 



180 THE ELLIPTIC INTEGRALS 

If the threads are wound in opposite directions on the reel, 
then R will describe a confocal ellipse, as at first ; but in this 
case the reel may be suppressed, and the thread merely made 
to slide round the ellipse, as in the theorems of Graves and 
Chasles. 

Moreover, it is not necessary that the tangents RP, RQ 
should proceed to the same ellipse, but to any two fixed con- 
focals, and the same theorems hold. 

If tangents R P t RQ , are drawn to the ellipse from any 
other point R on the confocal hyperbola RR, forming with RP, 
RQ the quadrilateral RrRY, then r, r lie on a confocal ellipse, 
by the preceding theorems ; and now a circle can be inscribed 
in this quadrilateral whose centre is at T, the point of concourse 
of the tangents to the confocals at R, T, R, r \ for TR, Tr, TR, 
Tr bisect the angles of the quadrilateral ; (Salmon, Conic 
Sections, 189). 

If R is brought up to L, the circle touches the ellipse at L ; 
so that the point of contact of the circle inscribed in the area 
bounded by two tangents and the ellipse is at the point where 
the confocal hyperbola through the point of intersection of the 
tangents cuts the ellipse. 

174. Putting u v = iv, or F<f> F\fr = Fy, 
then when v = Q and Q is at B, u = iu and P is at G where 
(j) = y suppose ; while R will come to 7) on the ellipse RD, where 
it is cut by the tangent at B. 

Now, since 

PR + RQ-arc PQ = BD + DG-&rc BG, 
or &rcPQ-wcBG = PR + RQ-BD-DG , 

therefore E$ E\}s Ey = a certain trigonometrical func 

tion of <p, \/s, y, which is found to be /c 2 sin <f> sin \fs sin y ; 
this is the Addition Theorem for the Second Elliptic Integral. 



-r? r>T>? ?f cosx^-cos^ 2 79 fsin0-sini//" /I 

For PR 2 = a 2 ! sin - ^ -- rf- J- + 6 2 ^ r-*r- -- TV~ COS r 
\ sm^-^) j \ Bin(0-V0 




so that PR = aA0 - . 



while BD = r -- -. 

Sin y 



OF THE SECOND AND THIRD KIND. 
Therefore, by 121, 



sin y 



= a {ooa cos \lr -f sin sin i/^Ay cos(0 \f/)} 









l-A 2 y . 

= a - - L sin sm \/r 
sin y 

= a/c 2 sin sin \fs sin y. 
In Jacobi s notation this is written 

E am uE&rnvE am(i& r), or Zu Zr Z(u v) 
= K 2 sn -it sn v sn(u v). 

175. Putting -y = w, and therefore i = Zw t then 

# am 2w 2E am w = /c 2 sn 2w sn 2/ w;, 
or changing w into Jie?, 

.Z? am iv 2E am Jtt; = /c 2 sn la sn 2 Jw; = sn w- ( 123). 

Then PjR + jRQ-arc P = BJD + DG-wc BG 



, , x l cni/j r, 

= a(l + an w) -- aE am i(; 
sn w 

sn w 1 dn 

\-asuw 



/ sn w j-, f \ o /sn w dn to \ 

= 2a( ^am4te;)=2a( ^ ^amivj : 

\1 + en i/j / \ en -kiv / 

and now en \w t or en ^(u v) = b//3, where fi=OK. 



176. A ready way of proving the Addition Theorem is to 
take the spherical triangle of Class II. , in which 
A = am v v B = am v 9 , C = am V B , 
wh ere i\ + v. 2 + 1 3 = 2/i , 

and to vary all the sides and angles, keeping K constant. 

Then dv 1 + dv z + dv s = 0, 

or dA /cos a + dB/cos b + cZ(7/cos c = 0, 

or cos 6 cos c . dA + cos c cos a . dB + cos a cos 6 . dC = 0, 
or (cos a sin 6 sin c cos J.)cLl + (cos 6 sin c sin a cos 
. + (cos c sin a sin 6 cos C)dC = 0, 
or cos ad A -f cos 6c?5+cos ccZO 
os J.cZ^ 

= K 2 d(siu A sin B sin (7). 



182 THE ELLIPTIC INTEGRALS 

Integrating, 

E(A ) + E(B) + E(C) - 2E= /c 2 sin A sin B sin 0, 

since y cos adA =/J(i - K *n*A)dA = E(A), 

o 

and v 2 = makes B = 0, and ^4 + (7= TT, or A"( 

In Jacobi s notation 

E am ^ + E am t 2 + ^ am v s 2E = /c 2 sn ^s 
or Zv 1 + Zv 2 -f Zi> 3 = /c 2 sn 

with ^ + v 2 + v 3 = 2if. 

With 



or Zu + Zv Z(i6 + v) = /c 2 sn u sn v sn (u + v). 

Fagnano s Theorems. 

177. The particular case of the -Addition Theorem, obtained 
by putting y = j7r, or u v = K, was discovered by Fagnano 
(1716), and leads to his theorems, namely, that if P, Q are two 
points on an ellipse of excentricity K, whose excentric angles 
0, \fs, measured from the minor axis, are such that 
A0 AI/A = K, or tan tan \{s = l//c = a/b, 
then the arc BP + arc BQ arc J.5 = /c 2 sin sin i/^, 
or arc BP arc J. Q = a /c 2 sin sin i/r = Ax /a ; 

x i x /: 2. a 2 

and then tan 2 tan V = ^^-__- 2 ^ = _, 

or K 2 x V 2 - ct 2 ^ 2 + ic /2 + a 4 - 0. 



On reference to tig. 23 it will be found that, if OF, OZ are 
the perpendiculars on the tangents at P and Q, then 
(i.) ^40^=0, AOY=^, 
(ii.) wcBP-sircAQ = PY=QZ=VQ-PT, - 

so that VZ=PT, and PFor QZ=K*xx /a , 
the tangents at P, Q meeting 0-4, 01? in T, F; 
(iii.) OP 2 -OQ 2 = OF 2 -0 2 ; (iv.) OY.OZ=ab. 
When P and Q coincide in F, then P is called Fagnano s 
point ; and then 

(i.) the arc BF arc AF= a-~b\ 

Va? 
- T-, 

(iii.) 



OF THE SECOND AND THIRD KIND. 



183 



(iv.) the tangents at P, Q intersect in R on the confocal 
hyperbola FED, through F, D, whose equation is 



(v.) the tangents at P and Q intersect in R on the confocal 
ellipse KDH, through K, D, H, whose equation is 



(vi.) PR-eL 
(vii.) the circle inscribed in the region bounded by A D, DB 

and the ellipse AB touches the ellipse at F; etc. 
The proof of these theorems is left as an exercise. 




o 



Fig. 23. 

178. Denoting the arc J.Pby s, the perpendicular OFon the 
tangent at P by p, the angle A T by \fr, then by Legendre s 
formula 

ds d? 



so that s + PY= /pd\[f ; 
and in the ellipse 

p = 
while 

P F= dp[d\fr = a/c 2 sin ^ cos i/r/ A\^ = a/c 2 sin sin \f, ; 
so that s + a/c 2 sin ^ sin ^ = afk\}sd\{s = aE\fs = arc 5Q, 

or arc J2Q arc AP = a/c 2 sin ^> sin i/r, 

as at first, in Fagnano s Theorem. 



184 THE ELLIPTIC INTEGRALS 

Confocal Ellipses and Hyperbolas. 
179. If we put 



then x = c sin cosh 0, y = ccos(J> sinh 9 ; 

T> 2 n 

so that --^ + -0 =C 2 
2 2 



c 2 

^ > 



the equations of a system of confocal ellipses and hyperbolas, 
since cosh 2 $ sinh 2 6) = sin 2 + cos 2 ^ = 1. 

T , n dx 2 dy 2 dx 2 dy 2 2/ , 2 . 0jN 

5^ + 4 5= d0 2+ ^ = c(cosh e ~ sin *> ; 
so that, in an ellipse BP, along which is constant, the 

arc BP = c/ / J(cosWO-sm 2 <j))d<p = aE</> 

as before, with a = c cosh 0, and the modulus equal to the 
excentricity sech 6. 

For the confocal hyperbola, along which < is constant, the 
arc is given by 



which can be expressed by elliptic integrals of the first and 
second kind, of Legendre s form. 
Putting 



the equation of the hyperbola is 



and now the coordinates of any point P on the hyperbola may 
be given by a cosec % b cot ^ ; and the tangent at P by 



and then amh 6 = JTT x, 

cosh 6 = cosec x, sinh = cot x, tanh = cos x, etc. 
The tangents at P, and at another point Q defined by 
will therefore meet at a point R, where 

a cosec x cot x cosec x cot x cos x cos x b cos x 
When we put 

v = am u, v = am v 

s\. * /\. 

the modular angle being 0, then as in 173 for the ellipse, 



OF THE SECOND AND THIRD KIND. 
SaCoCZ, Co Cn ifu - 1 1 ) 



185 



sn 



v) dn J(it v) 



_ 

b s 1 cZ 1 s 2 (i 2 SjcZg sn J (it + v ) d n J (it v) 

and therefore, eliminating en ^(u v) and dn i(u v), 



a en 



/csn 



/csn 



Ksn(tt+f) 



where a = 

and 

so that J describes a confocal ellipse, when u + v is constant. 




Fig. 24. 

180. By putting u-\-v = K, we obtain theorems for the hyper 
bola (fig. 24) analogous to Fagnano s theorems for the ellipse. 

Now ( 123) a 
or a 2 

and j describes the ellipse -FT), whose equation is 

o o 

x 



which will intersect the hyperbola in a point F, the analogue 

of Fagnano s point on the ellipse, the coordinates of which are 

c sin 0^/(l + cos 0), c(cos )*. 



186 THE ELLIPTIC INTEGRALS 

Now, as in 57, with 

X = am U, x = am v, and 



and cot x cot % = K = cos 0, 

or sinhflsinh 6 = K, 

and if a;, y and a? , ^ denote the coordinates of P and Q, 

a; = a cosec x = a Ax7 cos x > ^ = a cosec x = ^Ax/cos x ; 

y = a cotx=a* tnx , 2/ = a cot x = a* tan x ; 
and thus yy = aY = C 2 cos 3 <. 

Drawing the perpendiculars Y, OZ from on the tangents 
at P, Q, and denoting the angles AOY, AOZ by o>, ; then 

cfcc w/6 2 
tan co = .p = ~r-2 = tan </> cos x = tan < tanh = sin sin x /Ax ; 

sin ft> = sin < sin X , cosft> = Ax / , sin w = sin sin x, cos = Ax- 
Now denoting OF, OZ by p, p , then 

_p = ^/(a^cos 2 ^ 6 2 sin 2 ft)) = c^sin 2 ^ sin 2 o>) = c sin <p cos x ; 
pp = C 2 sin 2 0cosxcosx = C 2 sin 2 ^c 
Making use of the formulas 

ds d 2 p dp 

-j-= T^> p, and PF= - 7 , 
c^oj dw 2 l da) 

then 

PF- arc AP = 



also P F= c sin co cos (o/^sin 2 ^ sin 2 co) 

= c tan x Ax = c/tan x AX 
= c cosh sinh 0/^/(cosh 2 9 - sin 2 ^). 

181. The arc AP of the hyperbola is now expressed in terms 
of an elliptic integral of the first and of the second kind ; we 
can however express the arc by means of two elliptic integrals 
of the second kind, or by two elliptic arcs by means of Lan- 
den s transformation ( 67). 

We shall find that if we put 

2\ or sin2ir 



sin2\/r , (1-f sin0)A(i/r, y) 

then tan y = -r Y . . , sec v = / 

A sin 



, 4sin0 , 

where y 2 = , 7 = 



OF THE SECOND AND THIRD KIND. 187 



_ 

n x ~ 0)A(Vr,y) ~ ( 

A , 1 + sin </> cos 2 



, 



so that 

l + sin 



^-sinX) ~ A x -. (l + sin0)A(^, y) 
cos a) + ^/(sin 2 ^ sinV) = AX + K cos x = (1 + sin 0)A(i/r, y) ; 



Integrating, 

(i 

and now the arc of the hyperbola 



182. If we put 
then we find ( 180) 



= = 

1 cos tan 2 ^ 1 cos 



sn = 



A / ^ > x __ l-(l-cos0)sin 2 x / _ A 2 x r +cos0 
- 

and 



_ 

Now, sin(2 x - f ) = X sin g 

as in Landen s second transformation ( 123); and 
(1 + cos 0)A( X)c/ - ( A 2 + cos 



= 2A x d x + 2 cos , - siu 
Integrating, 



(1 +cos 0)^(f , X) = 2^x / + 2 cos <pF x - sin 2 ^ sin x cos xV^x j 
and the arc J.P can be expressed by means of E^ and E(, X). 

When x = x / = am i^ then ^^i^r; 
also ( 175) 2Y = ^(/c) + 1 - cos 0, while 2^ = -2" ; 
so that (1 + K )^(X) = ^(/c) + K K. 



188 THE ELLIPTIC INTEGRALS 

183. The following theorems, analogous to those of 177, 
can easily be proved by the student : 

(i.) The difference between the infinite asymptote DT and 

the infinite arc FT is equal to AD arc AF\ so that 

the difference between the infinite asymptote OT and 

the infinite arc AT is equal to OD + AD -2 &rcAF , 

(ii.) the coordinates of F are (c + &),/{(c-&)/c}, *J(V/c); 

and the tangent FK=AD = b, KG = c; 

(iii.) the tangents at P, Q intersect in R on the confocal 
ellipse through F, whose equation is 



~ 



and the tangents at P , Q intersect in R on the con- 
focal hyperbola through D and K, whose equation is 



= c; 

c a a 

(iv.) PR - arc PF=QR- arc QJP ; 

(v.) P R + R Q- arc P Q is constant; 

(vi.) the circle inscribed in the region bounded by the 
straight line AD, the asymptote DT and the hyper 
bola AQ touches the hyperbola at F\ 

PT.QV = FK 2 , PY.QZ=c 2 , 
Qv-PT=QZ, or vZ = PT, 



sin x cos x cos x sin x cos x cos x 

184. The geometrical theorems of 173 for the ellipse hold 
with slight modification for the mechanical description of con- 
focal ellipses and hyperbolas from a fixed hyperbola. 

The threads from the reel must be led round distant points 
on the hyperbola APQ (fig. 24) and be wrapped on the curve ; 
and now, starting from F, the confocal ellipse FED will be 
described, if the threads are led off in the same direction. 

At D, one thread DT must be supposed of infinite length ; 
and, beyond D on the ellipse FD, the thread DT must be trans 
ferred to the other branch of the hyperbola. 

By making the threads come off the reel in opposite direc 
tions, the confocal hyperbola DK can be described, starting 
from D or any other point R. 



OF THE SECOND AND THIRD KIND. 139 

185. The integration of the functions of 77 can now be 
expressed by means of the elliptic functions, and of the function 
E am u, defined by 

E am u =fdii 2 udu. 

o 

Then ficstfudu = u E am u 

o 

jK 2 GJ^udu = E am u K 2 u. 
o 

To integrate a reciprocal function, for instance nd 2 u, we 
notice that 

-y- ^ log dn u = K 2 nd 2 u dn 2 u, 

Cf/U" 

so that fK 2 nd 2 udu = E am u /c 2 sn u en u/dn u ; 

o 
and so on. 

Again, since cd 2 i6 = sn 2 (j5T u), 

jK 2 cd 2 udu = u jdu 2 (K u)du 



= u E + E am ( K u) 
= u E am u + /c 2 sn u en u/dn u : 
and since K 2 nd 2 u = dn 2 ^ u), 

fK 2 nd 2 udu = EE &m(Ku) 

= E am u /c 2 sn u en u/dn u, 
as before. 

In Problem III, 86, we find 
dt 



and 7i* =&&QdQ = 0- E am + sn dn 0/cn 0. 



EXAMPLES. 
1. Prove that the area of the Cassinian 



is 2 / (6 4 a 4 sin 2 <) 2 cZ0, if b > a ; 

o 

/~ iT 4 

or 27 (a^ o*sm 2 (h)~^b 4 cos~cbdd), if a > 6. 



190 THE ELLIPTIC INTEGRALS 

2. Rectify, by means of elliptic arcs (pointing out the 
geometrical connexion), 

(i.) y/b = sin x/a, cos x/a, cosh x/a, dn x/a, en x/a, sn x/a, . . . ; 
(ii.) r = bcos(bO/a) or acos(a9/b), the pedals of an epi- or 

hypo-cycloid ; 

(iii.) rcos(bO/a) = b, or rcosh(b6/a) = b, Cotes s spirals; 
(iv.) the Iima9on r = a + bcos0, the trochoid, and the epi- 

and hypo-trochoids. 

3. Express x as a function of s in the Elastica of 97. 
Prove that if the ordinate is made equal to p, the perpendic 

ular on the tangent from the centre of an ellipse or hyperbola, 
and if the abscissa is made equal to the &rcAPPY, the 
curve will be an Elastica (Maclaurin, Fluxions, 1742.) 



,, z . - K v 

-i. Prove that (1 K 2 )-r^ + -5 -- K=Q: 
x die K OK 

d*E \- dE 

(1 *rhrx+~ j- +E =0. 

cue K OK 

Change the independent variable in these differential equa 
tions from K to k, 6, or u, where 

K = ^k = sin = tanh u ; 
and reduce the resulting equations to the canonical form 



7 o 

y dx 2 
Solve the differential equations in which 

1 _ // 

I=-7T9jj*> cosec 2 2$, cosech 2 2u, 
4Ar/C z 

(Glaisher, Q. Jl Jf., XX., p. 313 ; Kleiber, Messenger, XVIII., 
p. 167.) 

5. Prove that, if u l -}-u 2 + u 3 -\-u^ 0, 



12S4 ll,. 3 x 4 

~ 




OF THE SECOND AND THIRD KIND. 

The Elliptic Integral of the Third Kind. 

186. We can now make a fresh start, and prove the Addition 
Theorem for the Zeta Function independently ; and then pro 
ceed to Jacobi s form of the Third Elliptic Integral. 
(Fundamenta Nova, 49; Glaisher, Proc. L.M.S. XVII. p. 153.) 

Multiplying formulas (3) and (6), 137, 

4/c 2 sn u en u dn u sn v en v dn v /1N 
,..(1) 
2 



(1-/C 2 S1 

and, integrating with respect to v, 

x n 2 en u dn u/sn it 



where C is the constant of integration, independent of v. 
To determine C, first put v = u\ then 

2 en u dn u/sn u 



so that, replacing E am u by Eu/K+Zu, 

, \ . TJ, \ ryo 2 en u dn tt/sn u 2 en u dn ulsn u 
Z(u+v)-f-Z(tt-v)-Z2tt= - = - H --- i - 5-^5 ^ 

1 /c z sn 4 u 1 rTsnnc. su*v 



n /csn 

1 



sn 2 u\ 

= K -sn(u + v)su(u-v)su2u .......... (2) 

Replacing u+v t u v, and 2u by u, v, and tt+t?, this 
becomes the formula given above, 176, 

Zu + Zv - Z(u -f- v) = /c 2 sn u sn v sn(u + v) ............. (2)* 

Again, put u = Q for the determination of 0; then 
C= 2Eu + 2 en u dn u/sn ^ ; 
and now 

2/c 2 sn u en u dn u sn 2 v 

(3), 



1 /c 2 sn 2 w, sn 2 t> 

another form of the Addition Equation of the Zeta Function, 
leading immediately to Jacobi s form of the Third Elliptic 
Integral, as required in 114. 

187. Integrating this equation (3) again with respect to v, and 
employing Jacobi s notation of 

TT/ \ r /Vsn u en u dn u sn 2 i; dv 
H(v, u) for / 
J 



1 

o 

where u is called the parameter, and v the argument, then 

IK 16 = vZu - 



192 THE ELLIPTIC INTEGRALS 

Jacob! now introduces a new function Qu, called the Theta 
Function, defined by 

\og~, 



or 



so that 
Now 



o 
/Z(u v)dv = log 



and 



,., 



so that the Third Elliptic Integral is expressed by Jacobi s 
Theta and Zeta Functions, the arguments being u and v, two 
in number only, and not three, n, K, </>, as in Legendre s form. 

188. Integrating equation (3) again with respect to u, 

f u f v 

/ 7{dn 2 (t + v} dn 2 (i6 v)}dv du = log(l 



o o 
or 



Q(u v) _, Qu 
-^-~ ~ 2 lo ~ = 



, ... 
log(l - 



or - = l-^n%sn 2 ^, ......... (6) 



a formula which takes the place of the Addition Theorem for 
the Theta Functions. 

For instance, putting u = v, 

e%u = (l- /c 2 sn%)e%/6 3 ................ (7) 

Interchanging the argument and parameter, u and v, then 



so that II(^, v)-II(v, ^) = uZv-?;Zu, ........................... (8) 

and Ii(v, u) is thus made to depend upon II(^, v). 



OF THE SECOND AND THIRD KIND. 193 

189. In Legendre s notation, II(7i, /c, 0) or simply 110, is 
employed to denote his Elliptic Integral of the Third Kind 





n being called Legendre s parameter ( 114) ; and with Jacobi s 



notation, H(, K, am u) = 



But Jacobi changes the notation, by putting n= /c 2 sn 2 a, 
and by calling a the parameter ; also by denoting the integral 
V 2 sn a en a dn a snhidu , -,-,- , 



o 
and not the integral 

du ,. , sn a H(u, a) 

2 ^ 5-, which equals u-\ 

/c 2 sn-a sn-n, en a dn a 

o 

In Legendre s notation, the Addition Equation of the elliptic 
integrals of the first kind 



leads to E<p + E\fr E/JL = /c 2 sin sin i/r sin /A, 

the Addition Theorem for the second elliptic integrals ; 
and now for Legendre s elliptic integrals of the third kind, 
the Addition Theorem is (Legendre, F. E. /., Chap. XVI.) 

TT , i TT ; TT 1 n*./a sin d> sin \!s sin /m 

H$ + H\k-U/uL= r tan- 1 - -^_ 

V a 1 + 71 71 COS <j> COS T/r COS /JL 

= -J-, tanh-i > /( - a)siD sin t SiPM , (9) 
^( a) 1 + n n cos cos \js cos /x 

according as a is positive or negative, where 



this can be verified by differentiation. 

This relation is very much simplified by the use of Jacobi s 
function II(u, a) ; and now with 



it becomes H(u, a) + II(v, a) II (u + v, a) = J log fi, 
where n_B(u-a)9(t;-a)e(u + T; + a) 

- ............. (1 



and 1 is capable of being expressed in a great variety of ways 
by means of the elliptic functions en, sn, dn of combinations 
of u, v, a. 

G.E.F. N 



194 THE ELLIPTIC INTEGRALS 



F 





/6Q - a)e(v - a)\ 2 _ 
~ 



80 ~J ~I- K %n*(;u-a)sn(v-a) 



00 ~) ~lif 

/ea9(u + 0-a)\ 2 9(M + fl)9(u + tt-2a) 
I 00 / ~l- K Wasn\u+v-a) 



90" 

( 188), so that (Fundamenta Nova, 54) 

Q 2 _ 1 /c 2 sn 2 (u + a)sn 2 (i> + a) 1 /c 2 sn 2 a sn 2 (t& -f 1; a) _ _ , 

sn 2 (u + v + a)" 



One of the simplest expressions, equivalent to that given 
above in (9) in Legendre s notation, is 

Q _ 1 it 2 snusn vsna sn(u-\-v a) ,- . 

l+/c 2 snt(/snvsnasn(u + v + a) " 

and a systematic collection of different forms of Q is given by 
Glaisher (Messenger of Mathematics, X.). 

190. According as Legendre s or (1 +TI)(! + 2 /^) is positive 
or negative, so his Integral of the Third Kind LT(X /c, 0) falls 
into one of two classes, the first called circular, the second 
logarithmic, or hyperbolic, as we shall call it. 

In the corresponding classification of Jacobi s form, the para 
meter a is imaginary or real; and it is remarkable that in 
dynamical problems, it is the circular form, with imaginary 
Jacobian parameter a, which is of almost invariable occurrence. 

When Legendre s 

a or (l+7i)(l+/c 2 /?i) 

is positive, and the corresponding Elliptic Integral of the Third 
Kind is circular, then Jacobi s parameter is imaginary; and 

(i.) with n positive, we must put n= /c 2 sn 2 m; 

(ii.) K 2 >n> 1, we must, according to 56, put 



as in 114 ; and now the integral is expressed by 

H.(u, id] or H(u, K+ib), 

involving Theta and Zeta functions of the imaginary arguments 
ia or K-\-ib ; for which there is no theorem, short of expansion, 
to express the result in a real form. 

We shall find however, in the applications, that this imagi 
nary form constitutes no real practical drawback. 



OF THE SECOND AND THIRD KIXD. 195 

Taking for example the result of 114, then, by (6) 188, 



with u = nt, and a = K+t iK ; while 





cm 



so that, by multiplication, 

(x + iy)(cos fjit i sin /mt), or p exp i(0 



-s/C 



AC J QuBa 
which, when resolved into its real and imaginary part, gives 
the vector of the herpolhode, or its coordinates with respect to 
axes resolving with constant angular velocity //. 

191. Take Jacobi s IL(u, a), and split up the quantity under 
the sign of integration into a quotient and partial fractions ; 
therefore 

Icnadnaf /"" du C du 

2 sn a 1^/1 K sn a sn u ,y 1 + K sn a si 
= u en a dn a/sn a + II(it, a) ; 
while 
1 cnadn a f f du f du 



:( f- 

( J 1 



2 sn a t J 1 K sn a sn u J 1 + /c sn a sn 16 J 
V en a dn a sn u , 



/csn 

c sn (a + u) J/c sn(a u) } du 
/c cn(a + u) dn 



_ 




Therefore, by addition and subtraction, 

en a dn a r du f en a dn a\ 

I - = u ZcH 

sna V 1 K sn a sn u \ sna/ 

o 

, 1^ Q(a u) dn(a u) /ccn(a 





2 t>(a + u)*dn(a+u)+/ccn(a+w) dna /ccna 

cnadna/" a lt / cnadnaN 

/r =su(ZaH 

sn a ^y 1 + /c sn a sn u sn a / 

. 1, 0(a 11) dn( ^() + /ccn(a u) dna /ccna 
& 



196 THE ELLIPTIC INTEGRALS 

192. Again, taking the formula (7), 137, 
sn 2 a 



= sn(a + tt)sn(a tt). 

Ksna snu 

and differentiating logarithmically with respect to a, 



Ksn sn 

_1 cn(a + u)dn(a+u) 1 cn(a u)dn(a u) 
~~2 sn(<x + u) 2 su(a-u) 

and then integrating with respect to u, 



snacnadnadu 1, sn(o. + i6) 

S n^-sn% = 2 10g 4^) E(W> ft 



a-w) 0(o tt) 

a + .) 
- ) .............. (14) 



introducing Jacobi s function Hu, called the Eta Function, 
defined by the equation (Fundamenta Nova, 61), 

snu = J_I^ 

^/K Bu" 

This form (14) and Jacobi s II(u, a) are the two forms of the 
hyperbolic integral of the third kind to which Legendre s form 
can be reduced for negative values of a. 

When > n > /c 2 , we put n = /c 2 sn 2 a, 

and obtain Jacobi s form TL(u, a) of (5). 

When 1 > n> oo,we put n = l/sn 2 a, 

and obtain the above form (14). 

This form again can be split up into partial fractions ; and 
a similar procedure shows that, since 

du __, snu , dnu cnu 

therefore, by equations (4) and (7), 137, 
cnadnasnudu 



f 



2 ,y 



sn (a + u) sn( u) , 



sn(a u) J sn (a + u) 



OF THE SECOND AND THIRD KIND. 197 

cn(g u) dng cng 



cn(g + u) dna + cng* 
. -, + cn(a + u) dng cng 
u) cn(g u) dna + cna ^ 
Therefore, by addition and subtraction of (14) and (16), 

/cnadnadu 
sn g sn u 
o 

i 11 0(g+u) dn(g + w)-j-cn(g + it) dng cng 

G(a u) dn(a u) cn(g u) dna-fcna* 

/cnadnadu 
sna+snu 



) en (a -h u) dn g + en g 

(a u) dn(g w) + cn(g ii) dn g en g* 

By means of equation (6), 188, and the formulas of 123, 
these relations may be written 

/cnadnadu 
sn a sn u 

o 

rj -i 2 ^(g + ii) sn ^g en ^(c 

U/jA-f- iO^ T^TT x 

U~^(g it) sn ^(a 

/cngdngfe 
sn g + sn u 

o 



The student may prove, by a similar procedure, that 



/ 



dnu-dna 



_ 1 
= 



//c 2 snacna(?-i6 _ , , 1 + dn(a it 
dnt + dna : 10g T + dH(a+i 
/snacnadna snucnudnu 7 , 
sn*a-sn tt 

/ 
y 



ciiu-cna 



= uZ _ 



198 THE ELLIPTIC INTEGRAL 

Eulers Pendulum. 

193. Consider for instance the rolling oscillations on a 
horizontal plane of a body with a cylindrical base, such as a 
rocking stone, or a cradle. 

Then the Principle of Energy, considering the line of contact 
as the instantaneous axis of rotation, leads to the equation 

JO 2 - 2ch cos + h 2 + k 2 )(d6/dt) = #&( vers a - vers 0) , 
where denotes the inclination to the vertical of the plane 
through the axis and the centre of gravity at any time t, a the 
extreme value of 0, c the radius of the cylindrical surface, h the 
distance of the C. G. from the axis of the cylinder, and k the 
radius of gyration about the parallel axis through the C. G. 

When c = 0, this equation reduces to ordinary pendulum 
oscillations, as in (3) 3 ; but in the general case we have the 
oscillations of what is sometimes called Euler s Pendulum. 

Th d? = {(c- 



4tgh cos 2 
and now, if we put 

tan J$ = tan J cos 0, 



dt 





on putting 7i 2 = r//c, and 

2 _ (c + fe) 2 + fe 2 . /2 _ (c-hY + k* 

~c 2 -2c/icosa + A 2 + /c 2S1 ~c 2 -2^cosa + ^ 2 + A: 2C( 

To reduce this to Jacobi s canonical form, put <j> = &mu, 
and sin 2 Ja = /c 2 sn 2 a ; then dn 2 a = cos 2 ia, 



- 
and sn 2 a = - , y^ -- , cn 2 a = 



, y^ , ., , . . 2 7g 

(c + /i) 2 + & 2 (c + /i) 2 + k 2 



dt ^snadna dn 2 u 

so that n = 2 



en a, 1 

_ sn a dn a _ 2/c 2 sn a en a dn a sn 2 u 
en a 1 /c 2 sn 2 

, _sn a dn a 

and 7i^=2 u 2II(u, a) 

en a 

while 



OF THE SECOND AND THIRD KIND. 199 

In the ordinary pendulum, where c = 0, this reduces, as 
in 8, to 



equivalent to 

sin|$ = sin Ja sn nt ; 



where n now denotes 

As another application of the Third Elliptic Integral the 
student may rectify the inverse (or pedal) of an ellipse or 
hyperbola, with respect to any point; examining the parti 
cular case when the point is the centre ; also the case of the 
Lemniscate, the inverse or pedal of a rectangular hyperbola, 
with respect to the centre (R. A. Roberts, Integral Calculus, 
p. 810). 



EXAMPLES. 



1. Prove that, if fc + //=!, 
(k c 



/ / 



and deduce Legendres relation of 171. 
2 f m f l _ k(y-x)dxdy 

o o 

/7w 



o 

x^W ^1 -. .7 7. 



.3 

J(l + Kx(^ K y (-x.-(y-l.- K ) 

(66). 

(y-x}dxdy _ = ^ 
-e B )^/(-4 ! .y-e r y-e. 2 .y-e 3 ) 

( 51). 
(y-x)dxdij _ 



f 
J 



( 47). 

(/3-a)( 7 -a}(S-a}(y-x)dxdy_ 



y |8 

7. Denoting K-E, K -E , E-K-K, E -^K by J, J , G, G f 
respectively (Glaisher, Q. J. M., XX.), prove that 

IK dK \ K f 2 (^dJ T dE\ 

-5 -- JK.J ) = - 1 JtLi - 7 -- J -j ] 

d K d K / K\ die die / 

( Jf dE r dJ \ 1 ( dG r ,dG\ 

= K (J -^ -- ~J l = - I VT^J -- ^"^T") 

V a/c a/c / K \ dK OK/ 



CHAPTER VII. 

ELLIPTIC INTEGRALS IN GENERAL, AND THEIR 
APPLICATIONS. 



?. The general algebraical function, the integral of which 
leads to elliptic integrals, is of the form 

S+TJX 
U+ VJX> 

where S, T, U, V are rational integral algebraical functions of 
x, and X is of the third or fourth degree in x. 
We first rationalize the denominator, so that 

S+TJX__(S+TJX)(U-VJX)_M N 1 
U+ vJX ~ U 2 -V*X ~D^D JX 9 

suppose ; and now the integration of the rational part M/D is 
effected by elementary methods, when it is resolved into its 
quotient and partial fractions. 

In the irrational part NjD^/X, the rational fraction N/D 
is also resolved, into a quotient, having a typical term x m , 
and into partial fractions, having typical terms 



By differentiation, we find that 



so that, integrating, and denoting Jx m dx/^/X by u m , 

x m - 3^/x = (m 1 )em m + 4(m f) bu m _ l + 6 (m 

+ 4(ra - %)du m _ 3 + (m - 3)eu m _ 4 , 

a formula of reduction by means of which the integral u m is 
made to depend ultimately on the integrals u 2 , u v and u . 



200 



ELLIPTIC INTEGRALS IN GENERAL. 201 

Similarly, by differentiation and integration, denoting 



by v m 
we can determine another formula of reduction, of the form 



-l = A V n + Bl n - 1 + Cl n _ -2 + Dv n . 3 + El n . 4, 



/ \n 

{Jb a) 

by means of which the integral v n is made to depend ultimately 
on the integrals v v v Qt V-^, and v_ 2 ; or rather, on v v u , u v u 2 ; 
since r and u are the same, and 

v -i = u i ~ w o u -2 = U 2 ~ 2ca 1 + a*w . 

By the various substitutions of Chapter II., u is reduced to 
Legendre s First Elliptic Integral, while at the same time the 
integrals u v u. 2 , and i\ are reduced to elliptic integrals of the 
Second and Third Kind. 

When x a is a factor of X, the substitution x a = l/y 
shows that v 1 becomes Jydyj^/T t where Fis a cubic function 
of y, and v l now reduces to the Second Elliptic Integral. 

But without carrying out this work in detail, now only of 
antiquarian interest, we adopt instead the Weierstrassian 
notation : and by means of the substitutions of the previous 
chapter we express x and ^/X rationally in terms of pu and 
p u ; so that the integration is reduced ultimately to that of 
A+Bp u with respect to u, A and B being rational functions 
of $11. 

195. We must at this stage introduce the functions 

f i& and <ru, 

the functions employed by Weierstrass, in conjunction with 
his function <pu. 

The function fit, called the zeta function, is defined by 

&= pu, or f&= Jpudu ; 
while the function a-u, called the sigma function, is defined by 



or log a-u =judu, (TIL = expy udu ; 



and thus - ^ = 9 u 
du 2 



202 ELLIPTIC INTEGRALS IN GENERAL, 

fi 

Taking the definition of s or pu in 50, 



expand in descending powers of s, and integrate ; then 



the * marking the place of a missing term in the expansion. 

Therefore, by Keversion of Series, since u 2 is a rational 
function of s, we obtain, in the neighbourhood of u = 0, 



To obtain further terms of the expansion, assume 

$> u = ^2+ * + c l u 2 + c^ + c 3 u Q +...+c n 
and since > 2 u = 4? 3 u 



we can obtain from the last equation a recurring formula for 
the determination of the coefficients c ; and as far as U B , 



The expansion of the zeta function is now 

fu = + -^ 3 -^!_ #2 V 

u^ 60 140 2 4 .3.5 2 .7 2 4 . 3. 5. 7. 11 

so that, defined more strictly, 



o 
Similarly we shall find, for the sigma function, 

f-f~/JI ; A I - |^ *-7 *J 3 ^_ IS 2 

~ * 2 4 .3.5 2 3 .3.5.7 2 9 .3 2 .5.7 2 7 .3 2 .5 2 .7.11 
so that, strictly defined, 

loc <ru = log u + / ( tu }du, or <jii = ^exp / \(u )du. 

J \* u/ v t/ V s u/ 



AND THEIR APPLICATIONS. 203 

Homogeneity. 

196. From considerations of homogeneity it follows, that if 
u is changed into u/m, and at the same time if g. 2 and g 3 are 
changed into ??i 4 <7 2 and m 6 (/ 3 , then s or <pu is changed into m 2 s 
or m 2 $w ; so that 

}- 1 ( Vj 

lit \ttir 

. \_ i-y fOL. 

and similarly 

_ 1 i(u 

a{u ; g.-,, g B ) = m <r ( ; 

At the same time the discriminant A becomes changed to 
m 12 A, but the absolute invariant J is left unchanged ( 53) ; 
we may in this manner alter the argument u proportionally ; 
for instance by taking m = i/(e 1 - c 3 ) we can make the argument 
the same as in the corresponding elliptic functions ( 51). 

When m is chosen so that m 12 A = l, or m=A"*,the elliptic 
integral is said to be normalised (Klein). 

Suppose, for instance, that g = 0, 

and m, m 2 are the imaginary cube roots of unity, J Ji^/3 ; 
then m 3 = l, and u/m = m z v,-, 

so that $p(m 2 it ; 0, g^ = m?<p(u ; 0, g 3 ), 
p(m u ; 0, g s ) = m f<u ; 0, g s ), 

1111(3 w iC ~-~ v) iit/tL O if1>~ / lJlf* 

<r(u 0, f/o) = m<r = m~v 

L/^/ /yyj />T1 

This is the simplest illustration of the theory of Complex 
Multiplication of Elliptic Functions, of which we shall make 
use hereafter ; the general theory is required in the integration 
of the equation 

Mdy dx 



for particular numerical values of g. 2 and </ 3 , when l/M is a 
complex number of the form a + ib^/n ; in this instance g. 2 = Q> 
and M is an imaginary cubic root of unity. 



204 ELLIPTIC INTEGRALS IN GENERAL, 

197. With the aid of these three functions of Weierstrass^ 
<pu, fit, and oru, it is possible to express any elliptic integral, 
and we can thus complete the problem left unfinished in 194. 

The function f u is analogous to Jacobi s Zeta function ; and 
with s = <@u } it may be defined by the relation 

& =J S -jji =/(4s 3 - 2 s - </ 3 ) - *s ds 



* 

Thus, for instance, from 153, with appropriate limits, 

. as-0-3 dx 

i 



\a /3 a y a 

where u=f^=. 

J sj -& 

To obtain the Addition Equation of the zeta function 
analogous to (2) and (3) of 186, take the formula (F) of 144, 

it ll, ft/? A ^ 



implying also the formula, obtained by changing the sign of v, 

ftt+r 

so that, by subtraction, 

, (u _ v) 

Integrating (a) with respect to v, 



where C, the arbitrary constant of integration, may be obtained 
by putting v = 0, when p; = oo ; so that 0= 2u, and 



An interchange of it and v gives 

-8-)+ 
so that, by addition, 



(y) 



the Addition Equation, analogous to (2*) 186. 



AND THEIR APPLICATIONS. 205 

With U + V + W = Q ) 

this may be written, analogous to 176, 



198. We can now take the function A+Bp u of 194, and 
suppose that A and B are resolved into their quotient and 
partial fractions. 

Writing p, p , p", ... for <pu and its successive derivatives, 
then the relations 

pt= 4^ 3 -#,p-(/ 3 

p"= W-to 

p" =l2pp\ etc., 

enable us to express the quotient or integral part of A + B p u 
in the form 

C = C 



Considering next a partial fraction of A + B y u of the form 



we replace a by pi 1 , and write the partial fraction in the form 



pu-pv 



All such partial fractions can thus be expressed by a series 
of terms, 



where the sum of the coefficients I is zero for each partial 
fraction, and therefore for the whole series ; so that 

? 1 + / 2 + / 3 +...=0. 

Again, by repeated differentiation of equations (/3) and (ft ) 
( 197), with respect to u or v, we obtain equations, such as 



by means of which partial fractions of the form 

P + Q& u .. P + Qy u 

,o, or generallv . . 

u w* 1 J * 



can be expressed by terms of the form ^(u + r), $>(u v), and 
by their derivatives ; as well as by terms of the form L and C. 



206 ELLIPTIC INTEGRALS IN GENERAL, 

Thus, finally, A+Bp u, or any rational function of <pu and 
p u, can always be expressed as the sum L + P of two series of 
terms, L = l(u - vj + l(u - v z ) + l B {(u - r s ) + . . . , 
where ^ + Z 2 _|_ g + . . . = Q, 

and P = c + 2m ^(u v) ; 

and now the integral can immediately be written down, in 
volving, in general, the sigma, zeta, and $ function, as well as 
its derivatives. 

When the sigma and zeta functions are absent, the integral 
is a function of @u and p u, and is not properly elliptic, but 
only algebraical. 

This method of integration is taken from Halphen s Fonc- 
tions Elliptiques, L, chap. vii. 

Halphen points out that to obtain the coefficients in the 
series of terms 

I flu -v)-}- m$(u -v) + m^\u -v} + m.$"(u - v) + . . . , 
corresponding to the same v, it is only necessary to take the 
coefficients of (u v)-\ (u-v)~ 2 , (u-v)~ 3 , ... in the expansion 
of A+B$ u in ascending powers of u v; the coefficient I 
being Cauchy s residue. 

199. Integrating (j3) with respect to v, then 

ff udv^ ^uy) ..... 

J <$U-<$V *(r(U-v) S lPl 



which may be considered a canonical form of the Third Elliptic 
Integral, in Weierstrass s notation. 
Thus, for instance, in 113, 






By integration of (y), with respect to u and v, 
I i u i v- -, 



/I ip u 
% pU 

/I W U Q VJ 
tZv = 
2 ^-^v 



AXD THEIR APPLICATIONS. 207 

either of which may be taken as a canonical form of the Third 
Elliptic Integral ; and also as illustrating the interchange of 
amplitude u and parameter v, as in the Jacobian Elliptic 
Integral of the Third Kind, TL(u, v), in 188. 

Or otherwise, interchanging u and v in (fa), or integrating (/3 )> 



so that, by addition of (fa) and (fa), 

. -.(<?) 



a form of the theorem of the interchange of amplitude and 
parameter, analogous to (8), 188. 

200. Integrating (ft) with respect to u, 






the fundamental formula is the use of Weierstrass s elliptic 
function, analogous to equation (6) of 188. 

As an application consider the herpolhode of 113 ; then 



(jUoV 

while #* = . l^ u + v ( e ~ t 

\<K*-t;) 

so that, in the curve described by H, 

fo + t 

a-ua-v 

while in the herpolhode described by P we must multiply this 
function by e iflt or cos /mt + i sin jmt. 
Putting u = v in (K), we obtain 






(7 (7 

This may be obtained by integration of the formula of 149, 
1 d 2 



208 ELLIPTIC INTEGRALS IN GENERAL, 

If u, v, w, x denote any four arguments, 

<r(u - v )<r( u + v )ar(w - x)a-(w + x) 



0, ............ (L) 

since it is of the form 



where U- V=- ^Ua^v^u - QV), etc. 

201. We notice that the Third Elliptic Integral can be 
expressed very simply as the logarithm of a function, so that 
we may write (y^ in the form 



uv ft/ - -v- ii V-*V ) ^ ) ) 

o 7 

i , . o-(u-\-v] } 
where ^C^, v) = -&-*** 

eru arv 

and </>(u, v) is called by Hermite a doubly periodic function of 
the second kind. 

Changing the sign of u, or v, 



(jU (TV 

so that <p(u, v)<p(u, v) = <@u $>v. 

202. Suppose @v = e lf e 2 , or e 3 ; then, according to 54, we 
can take v = u> v ^ + 0)3, or w 3 , to correspond; and now 

<p v = 0, and log (j>(u, v) = % log ($>u pv) ; 
so that 

0(u, wj) = <f>(u, - Wl ) = J($>u - ej, etc. ; 

and </>(u, v) is an elliptic function for these values of v. 
We may thus put 

, or ^, 



where a-.u denotes 

Similarly, 



where 



Also ^>% = - 2^/($>u -e r pu-e. 2 . <pu -e 3 ) = - Z^u cr. 2 u ar 3 u/a*ii, 
and ( 200) 



AND THEIR APPLICATIONS. 9Q9 

Denoting by a, /3, y the three numbers 1, 2, 3, taken in any 
order, then the relation 



gives, by a combination of the expansions of o~u and <pu in 195, 



so that o- a u is an even function of u, and unaffected by Homo 
geneity ( 196). 
Thus, for instance, from ex. 9, p. 174, 

>2u - e a 



The symbol ^ is employed to denote fa) a , so that r\ is 
the analogue of Legendre s E of 77. 

With positive discriminant A ( 53), we find (exs. 4, o, p. 199), 



and with negative A ( 62), 



formulas analogous to Legendre s relation of 171. 

203. Denoting $m, pv, pie; by x t y, z, then ( 165) if 

u + v + w = 0, 
(x^y + z}(^xyz-g z ) = (yz + zx + xy + lg^ ................. (I.) 

Denoting also (x ea)(y e a }(z e a ) by s a 2 , then since 



a + (a; 
__zzx x 2x ze 



by means of (I) ; and this is of the form A+Be a , so that 

(e, - g s )s i + 3 - ei )s. 2 + (c t - e 2 )s 3 = ; 
or (e z - e 3 V 1 uor 1 i o-^+(e 3 - e 1 )o- 2 ito- 2 7;o- 2 w + (e 1 - 62)0-3160-3^0-3^ = 0, 



snce 



(W. Burnside, Messenger of Mathematics, Oct. 1891.) 
As an exercise the student may prove that, with 

u + v + iv + x = 0, 
(e. 2 e a )criU (r-^v 



2 - 3 a - 11 - 

the analogue, in Weierstrass s notation, to Cayley s theorem, 
given in ex. 1, ii., p. 140. 

G.E.F. o 



210 ELLIPTIC INTEGRALS IN GENERAL, 

204. The solution of Lamp s differential equation, which may 
be written in Weierstrass s notation 



.(1) 



is given, when n = l, by the function 0(u, v) of 201. 
For, differentiating < logarithmically with respect to u, 

1 d(b 1 & u & v ,. N c c 
-- 5*-=5- - =(u + v) tu tv, 

$ du 2 ^u-^^ 

and differentiating again, 

1 d 2 <f> I d<{> 2 , 

<j>du*-fdu* = 
so that 



du 2 4 



Lame s differential equation, with n = l, and h = $v. 
The general solution of 



is therefore 

y = C<j>(u, v) + C (f>(u, v), or C<t>(u, v) + C <p( u, v). 
When h or @v = e lt e 2 , or e 3 , the solution is one of Lamp s 
functions, as in 202. 

One solution is now *J($u e a ), where a = l, 2, or 3; 
the other being 

{ f (u + co a ) e a u} /J^u e a ) t 

as may be verified by differentiation, or determined indepen 
dently from a knowledge of the particular solution ^/(<@u e a ). 

205. The revolving chain, resumed. 

We are now able to complete the solution ( 80) of the 
tortuous revolving chain, by obtaining an analytical expression 
for its projection on a plane perpendicular to the axis of 
revolution. 

Putting y = r cos i/r, z = r sin i/r, 

then we have found in 80, p. 70, that, when the notation of 
Legendre and Jacobi is employed, 

d^_H H/T 

dx ~ Tr* 



AND THEIR APPLICATIONS. 211 

which, on putting u = Kx/a, and 



so that, with K 2 = (6 2 - c 2 )/(cZ 2 - c 2 ), 

sn 2 v = - (d 2 - c 2 )/c 2 , cn 2 ^ = c? 2 /c 2 , 

i cZi\/r cn v dn v/sn v 

becomes ~f- = - - ^^ -- , 

ft i& 1 trSDrli sn v 

, , . , cn t dn i; ,-,, x ,, x 

so that i\/s=u- --- IL(u, v) ...................... (1) 

Since sn 2 y is negative, we may, by (67) 73, put v = t iK , 
where t is a real proper fraction. 
Now r = 



=ceo 



/ 

\ 



-,., .$ /9(u-fv) / cnvdnv \ 

while e l V=J 7 -- (exp --- -- Zv)u; 

\6(u y) snv / 

j.v w^ 0(^ + v) / cn v dn v \ 

so that y + iz = cOQ -^^ exp -- -- Zv)u:....(3) 

016 Ot; sn v 

which, when resolved into its real and imaginary part, will 
give y and as functions of u or Kxja, and thus represent the 
equation of the chain. 

y 

^ 206. The procedure is more rapid with Weierstrass s notation. 
Writing 2/ 2 + z 2 = r 2 , we have found that ( 80) 

( r 7.,2\2 /4, / ,2 
=^o--^+^-c), 

so that we may put 

- S w), ..................................... (1) 



j j J.V j. 
provided that ~-i-= 

and g 2 , g% are suitably chosen. 

Since v is the value of u which makes r 2 vanish, therefore 



the value of (dr 2 /dx) 2 when r 2 = ( 80) ; so that 

^ 2 y=: -16lT 2 /?i% 2 ^ 6 , ............................... (2) 

and <p v is therefore a pure imaginary, which we take to be 
negative imaginary, so that v = t w 3 ( 54). 

Now d = H_ dx^ -IE 1 )&v_ 

dn Tr 2 du n-wJ& pu pv %>u-$>v 



212 ELLIPTIC INTEGRALS IN GENERAL, 



or = --- = b&v + u) + tf(v-u)-tv ...... (3) 

du %>upv Jiv 

from (ft) ( 197) ; so that 



(i> tt) 




while r - k = ................... (5) 

- 2 * 

and + fe= fc 



crVcrU 

= Jc</>(u, v), 
y-iz = k<f>(u,-v), .................................. (6) 

giving the form of the chain. 

For a revolving chain fixed at two points, we must have r 2 
restricted to lie between positive values, 6 2 and c 2 , and therefore 
$u must be restricted to lie between e 2 and e 3 ; so that with 



du/dx constant, we must put u = x 

For a chain attracted to the axis with intensity proportional 
to the distance, and thus taking up a form of minimum 
moment of inertia, we have u = xcvja ; and now pu can become 
infinite, and the chain reach to infinite distance. 

In this and other mechanical problems, the parameter of the 
elliptic integral of the third kind is almost always imaginary ; 
the apparent awkwardness of this imaginary parameter is 
removed when we proceed to express the vector y-\-iz by a 
doubly periodic function of the second kind $(11,, v), whose 
logarithm is the elliptic integral of the third kind ; and thence 
determine y and z theoretically by resolving </>(u, v) into its 
real and imaginary part. 

Familiar instances of the same procedure are met with in 
Elementary Mathematics ; thus 

x + iy = c cos(nt + ia), or c cosh(nt + i/3), 
will represent elliptic or hyperbolic motion about the centre. 

Generally, with x -f iy z, X+iY=Z=F z: then 



will give the motion of a particle of unit mass under component 
forces (X, F). (Lecornu, Comptes Rendus, t. 101, p. 1244.) 



AND THEIR APPLICATIONS. 21:3 


207. The Tortuous Elastica. 

A procedure, similar to that just employed for the revolving 
chain, will show that the equation of the curve assumed by 
a round wire of uniform flexibility in all directions can be 
expressed by the equation 

y + iz = k<l>(u, v) 
and z = ku + yii, 

where u = swjc + w 3 , 

s denoting the length of an arc of the wire, and 2c the length 
of a complete wave. 

(Proc. London Math. Society, XVIIL, p. 277.) 
The elastic wire differs thus from the revolving chain in 
having it = so^/c + w s , instead of tt=a^ 1 /a-hw 8 ( 97). 

To establish these equations, take the axis Ox as the axis of 
the applied wrench, consisting of a force A^ along Ox and 
a couple L in a plane perpendicular to Ox ; denote the tor- 
sional couple about the tangent at any point by G, and the 
flexural rigidity of the wire by B. 

Then the component couples of resilience about the axes 
Ox, Oy, Oz are taken to be 

B(y z"-y"z), B(z x" -z"x ), B(x y"-x"y ) 
the accents denoting differentiation with respect to the arc s ; 
the equations of equilibrium are therefore 

B(yz"-y"z }=Gx+L (1) 

B(z x"-z"x) = Gy + Xz (2) 

B(xy"-x"y }=Gz-Xy (3) 

(Binet and Wantzel, Comptes Rendus, 1844). 
Differentiating each equation with respect to s, multiplying 
respectively by x, y, z, and adding, gives 

G = ; so that G is constant. 
Multiply equations (1), (2), (3) by x, y, z\ and add ; then 

-r L* "^- X(yz - y z) = 0, 

so that yz y z = r 2 d\fs/ds = G/X, a constant ; 

and yz" y"z = 0. 

Again, multiplying (2) by y, (3) by ~, and adding, gives 

Bx\yz - y z} - Bx (yz" - y"z) = G(yy f + zz \ 
or Bx" = X(yy + zz \ 

so that, integrating, Bx = ^X(y- + 2 ) + H. 



214 ELLIPTIC INTEGRALS IN GENERAL, 

Then 



= 2X(Bx -H)(I-x 2 )-G 2 , 

a cubic function of x ; so that, by inversion of the elliptic 
integral, x or y 2 +z 2 is an elliptic function of the arc s, which 
may be written 

y 2 + ^ = & 8 (pa,-pw), ........................... (4) 

or Bx 

. , , du 

provided A 

dx 
and now du 



also = iG ds = 2iBG 1 = frfa 

^ 23 



By KirchhofFs Kinetic Analogue, it follows that the axis of 
a Spherical Pendulum, Gyrostat, or Top can be made to follow 
in direction the tangent of a certain Tortuous Elastica, when 
the point of contact of the tangent on the elastica moves with 
constant velocity ; so that, if x, y, z are the coordinates of a 
point fixed in the axis of the Gyrostat, and Ox is vertical, 

7 d o-(u + a)) , . . 
= k- r - -exp(X MU, 

du cru o-a) 



where now u = ^ + o) 3 , 

and 2 1 /w is the period of the oscillations of the Top, or Spheri 

cal Pendulum. 

The Spherical Pendulum and the Top. 

208. To prove these formulas independently for the spheri 
cal pendulum, let the weight of the bob be W lb., and let the 
tension of the thread be a force of Nl W poundals ; then the 
equations of motion are, with the axis of x drawn vertically 
dovmwards, 

d 2 x d 2 d 2 z A /1X 

0; ......... (1) 



subject to the condition, I denoting the length of the thread, 



AND THEIR APPLICATIONS. 215 

The equation of energy is 



(2) 
while yz yz = /^, a constant ................ (3) 

Now, xic + T/i/ + 22 4- ^ 2 = ##, 

so that JV7 2 = </ + x 2 + # 2 + z 2 = g(3x + 2c) ; 
thus giving the tension of the thread. 

Hermite writes (Sur quelques applications des fonctions 
elliptiques, 1885) 

(y + iz)(y - iz) = yy + zz- i(yz - yz) 

= xxih, 
so that the norm of each side is 

(y 2 +z*)(y z +z*) = xW+h\ 
Then 

(V-x z ){2g(x + c)-x 2 } =xW+h?, 
or I 2 x 2 = 2g(x + c)(l 2 - x 2 ) - h 2 



so that x is a simple elliptic function of t, which we may write 

x = k(pv-&u), ........................ (4) 

where u = nt+a)^ for ^16 to lie between e. 2 and e s . 

Then l 2 k-,t Y 2 ii = 2gJ^u - pvf - 2gck 2 (pu - pv)* 

- pi ) + 2gd* - h 2 



provided n 2 = ^gkjl 2 , and py = Jc/& ; 

while ^ 2 and g 3 are suitably chosen. 

The value of y v is found by noticing that x = when u v; 
and thus l 2 k 2 nY 2 v = 2gcl 2 - h 2 , 

Now Hermite writes 



d 2 . 

j-5(3/2f= -r- 9 = -- 1- = , 

^ 2V< c/it 2 grfc A; 

Lame s differential equation for n = 2, with ^=6|w. 

The formal solution of this equation is reserved for the 
present; but it can be inferred for this case by taking the 
equation (3) and writing it 
^ = h 
du n(y 2 + z 2 ) 
di\ts_ ih/n _^i 



216 ELLIPTIC INTEGRALS IN GENERAL, 

We now put 



......... (6) 

and since I 2 x* = h z , when x = I, or when u = a, or 6, therefore 

79 



With & positive, and 06>0u>0a, we take p a negative 
imaginary, and 6 = a positive imaginary, so that ( 54), 
, where p and q are real proper fractions. 



Then = - +s_. ... (7) 

au 016 0a <pb $u 

and integrating, by equation (/3), 199, 

i 11 <r( U + a) . ., cr(6 + u) c7 /Q \ 

V =l 1 g^=r ) -f+i 1 8-^-( 8 > 

Now 
while 



j.i , 7 ,. - 

so that v + ^ = ^ - exp( - fa - 

era 0-6 9% 

-,cr(u a)a-(b u 



/nx 
(9) 



thus giving the solution of Lame s differential equation for n 2. 

209. It is interesting to verify that these values of y + iz 
and y iz are solutions of Lame s equation for n = 2. 

Denoting y-\-iz by 0, and differentiating logarithmically, 



and differentiating again, 



1 p u-p a 1 p b-p u. 

2 u wt 2 <b <u 



l/0 i6 a\ 2 . 1 <QU a 6 0V l/0 6 /r i 

= . I -5 ) ""~ 9 ^^ 1 + 4^7; 

-K6+^) 




AND THEIR APPLICATIONS. 217 

u &a p 6 p u 

a- T 

2 pu pa po*-fne. 

But with p a = p 6, 

tt p a p 6 & u 

- - \ 

- pit) 

-rt = 6pit -{- 3pa + 3p6, 

<h CiU 

Lame s differential equation for ?i = 2, with 7t = 3pa + 3p6, in 
place of the previous value of h = 6pv. 

From KirchhofF s Kinetic Analogue in 207 we may put 

-, N 
ex P( ft> tyu 



era orb cr 
d I 



where X = f (a + b) fa f 6. 

With p (a - 6) = p a = p 6, 

therefore f (a 6) = fa f 6 ; 

and, changing the sign of a, 

P7 cZ 

exp(ca Co)u=-j-A(tt, a 
^ 



7 9 
o-a 0-6 a- 2 u 



(Halphen, F. E., I., p. 230.) 



210. In the slightly more general case of the motion of the 
Top, we shall find it convenient to draw the axis Ox vertically 
upwards, and to call the angle which the axis OC of the 
top makes with the vertical Ox. 

Then, from the principles of the Conservation of Energy and 

Momentum, we obtain the equations (Routh, Rigid Dynamics) 

i A (dO/dty + \A sm 2 0(aV/cZ0 2 = Wg(c - h cos 0) , ...... ( 1 ) 

^sin 2 0(^/^0 + ^ cos0=^, ...................... (2) 

where r denotes the constant angular velocity of the top about 
its axis of figure OC, d\frjdt the angular velocity of the verti 
cal plane through Ox and OC, h the distance of the centre of 
gravity G from 0, W Ib. the weight of the top, and C, A 
its moments of inertia about the axis of figure OC } and about 
any axis through at right angles to OC. 



218 ELLIPTIC INTEGRALS IN GENERAL, 

Putting A/Wh = l=OP, as in the simple pendulum, then 
P is the centre of oscillation for plane vibrations. 

The elimination of d\fsldt between equations (1) and (2) gives 



= (/(cos cos a)(cos cos /3)(cos d), ...... (3) 

suppose ; the inclination of the axis of the top to the vertical 
being supposed to oscillate between a and /3, 

a > > /3, or cos a < cos < cos /3 < d. 
Guided by equation (17), p. 37, we put 

cos = cos a cos 2 < + cos /3 sin 2 <, 
cos cos a = (cos /3 cos a)sin 2 <, 
cos/3 cos 6 = (cos /3 cos a)cos 2 < ; ................ (4) 

and therefore, 



= o j{d cos a (cos/3 eosa)sin 2 </>} 



, 9 cos 8 cos a , 9 d cos ^8 

where /r = -- s " K ^ 
d cos a a cos a 

and in- 2 = J </(c? cos a). 

Now we may put < = am nt, and 

cos = cos a cn 2 ?i + cos /3 sn 2 7i, ................ (5) 

so that the projection on the vertical Ox of the motion of a 
point on OC resembles ordinary plane pendulum motion. 

When d = 1 and cos a= 1, then 



G and Cr vanish, and the oscillations are in a vertical plane. 
But, in the general state of motion, 



._ 
dt = "sin 

= 1 G + Cr 1 0- 
2 



1 G^ + Or 1 _ G-Or _ 

2 l-cosa-(cos/3-cosa)sn 2 ft 



~2 

so that \/s is expressed by two Third Elliptic Integrals. 



AND THEIR APPLICATIONS. 219 

Putting cos#= 1 in equation (3), show that 



= 



= 2(l +cos a 



cos 



= 2(l -cos a)(l -cos /3)((Z- 1), 

i 

while, in accordance with Jacobi s notation, we put 



cos 3 cos a 9 cos / cos a . 
= 



, 

1 + cosa 1 cos a 

so that, finally, with u = nt, we find 



en t\n vsn. v t . en 



and, as in the spherical pendulum ( 208), we take 

v l = ipK , r. 2 = K+ iqK , 
where p and q are real proper fractions. 

In the Weierstrassian notation, we put, as in (6), 208, 

1 + cos = k($ni pa), 1 cos 6 = 
and thence ( 22-i) c - h cos = hk{p(a + b)- 



^1-1- AT, V A , 

We thus obtain -5-*-= --- 1 

du pu $>a 

but now the relation p a= p & holds only when 0;- = 0, or 
when the motion of the top is comparable with that of the 
spherical pendulum ; on the other hand, the relation p a = $ b 
implies that G = 0. 

The Kinetic Analogue of the Top with the Tortuous Elastica 
( 207) is obtained by putting 

a + b = a), and X = f(a + 6) a f&. 

In the Steady Motion of the Top, a = /3, K = 0, K= JTT ; 
and the elliptic functions degenerate into circular functions. 

We thus obtain the condition for the steady motion, and the 
period of the small oscillations, given in Ifoutb s Rigid Dynamics. 

211. A similar procedure will solve the general equations 
of motion of a solid figure of revolution, moving under no 
forces through an infinitely extended incompressible friction- 
less liquid ; the work will be found in Appendix III. of 
Basset s Hydrodynamics, vol. I ; also in Halphen s Fonctions 
elliptiques, II., chap. IV. The problem is of practical interest 
from its bearing upon the determination of the amount of spin 
requisite to secure the stability of an elongated projectile. 

(Proceedings, Royal Artillery Institution, 1879.) 



220 ELLIPTIC INTEGRALS IN GENERAL, 

212. We again resume the consideration of the motion of a 
body under no forces, first mentioned in 32, as affording a 
good practical illustration of the necessity for the introduction 
of various analytical theorems of Elliptic Functions. 

Geometrical Representation of the Motion of a Body under 
No Forces, according to MacCullagh, Siacci, and Gebbia. 

Quadrics concyclic with the moinental ellipsoid, that is, 
having the same circular sections, are given by (Smith, Solid 
Geometry, 170) 

(A -H)x 2 + (B- H)y 2 + (C- H)z 2 = Dk 2 ; 

and now, if we produce the instantaneous axis of rotation OP 
to meet the concyclic quadric in P , and denote OP by R, 

(A -H)p* + (B- H)q 2 + (C- H)r 2 = DAV/E 2 , 
while Ap* + Bq 2 + Cr^Dh 

so that, by subtraction, 



M 2 R 2 J C * R 2 R 2 ~ D 
Along the polhode, R h sec 6, where denotes the angle 
between the instantaneous axis OP and the fixed axis of 
resultant angular momentum 00 ; and then 

W = COS 2 6 _ H (l} 

the polar equation of a quadric surface of revolution. 

Since R 2 is less than A 2 sec 2 $ for all points adjacent to P on 
the momental ellipsoid, therefore in the concyclic quadric 

1 . cos 2 H 

-f^ 9 is greater than 



except at the point P , and therefore the concyclic quadric 
touches this quadric surface of revolution at P and rolls 
upon it during the motion. 

We may also take concyclic quadrics, given by 



and now __ C0 ^, ..................... (2) 

the polar equation of a quadric of revolution. 

In particular, if H = D, then .K sin 6 = h, the polar equation 
of a cylinder of revolution, outside which this concyclic hyper- 
boloid rolls during the motion (Siacci, In memoriam D. 
Cheliniy Collectanea mathematica, 1881.) 



AXD THEIR APPLICATIONS. 221 

213. By reciprocation of these theorems, we prove Mac- 
Cullagh s theorem, " that the ellipsoid of gyration, 

^ , f, z*_ I 
A^B^C M* 

always moves in contact with two fixed points on the axis of 
resultant angular momentum, equidistant from the centre " ; 
and we also deduce Gebbia s extension of MacCullagh s theorem, 
that " conf ocals of the ellipsoid of gyration, the polar recipro 
cals of the concyclic ellipsoids of the momental ellipsoid, slide 
without rolling on fixed quadric surfaces of revolution." 

In particular, the polar reciprocal of Siacci s cylinder of 
revolution is a circle, upon which a certain confocal to the 
ellipsoid of gyration slides without rolling. 

Geometrical Representation of the Motion, according to 
Sylvester, Darboux, omd Mannheim. 

214. In Sylvester s splendid generalization of Poinsot s re 
presentation of the motion of the body, it is proved that a 
confocal to the momental ellipsoid rolls upon a plane per 
pendicular to the axis of resultant angular momentum OC at 
a constant distance from 0, which plane rotates about OC with 
constant angular velocity, and therefore gives a geometrical 
representation of the time. (Phil. Trans., 1866.) 

The proof of this theorem depends upon tw r o geometrical 
propositions, in connexion with confocal quadric surfaces 

(i.) "The locus of the pole of a fixed tangent plane to a 
quadric surface, with respect to any confocal, is the normal to 
the first surface ; " 

(ii.) " the difference of the squares of the perpendiculars from 
the centre on two parallel tangent planes of two confocals is 
constant and equal to the difference of the squares of the 
corresponding semi-axes." 

Thus, in fig. 25, if OP is a surface confocal with the 
momental ellipsoid OP, then Q, the pole of the invariable 
plane CP with respect to the surface OP , will lie in the 
normal PQ to the momental ellipsoid at P ; while the surface 
OP will touch a plane C P , parallel to the invariable plane 
CP, and such that OC - = OC 2 -\~, X 2 denoting the difference 
of the squares of corresponding semi-axes of the confocals. 



222 



ELLIPTIC INTEGRALS IN GENERAL, 



Since C is a fixed point during the motion of the body, 
therefore C" is also fixed. 

Drawing the plane QL through Q, parallel to the invariable 
plane, and denoting OC by h, as before ; then since Q is the 
pole of OP, 

OQ.OV=OP" 2 , or OL.OC= 
so that OL = h- \ 2 /h, LC= 




Fig. 25. 

Again, denoting as before ( 104) by /JL the constant com 
ponent of the angular velocity of the body about OC, so 
that the resultant angular velocity of the body about OP is 
m cosec OPC, then the velocity of the point P in the body is 

p cosec OPC . OP . sin POP = ^.P V, 

where V is the point in which the line OP cuts the plane C P . 
Therefore the angular velocity of P about the invariable 

r nn . p v py PQ A 2 

line OU is /x (7 7 P 7 = M OP =M 00 :=/x /t 2 

a constant ; so that if the surface OP rolls without slipping 
on the plane C P , this plane must revolve about OC with 
constant angular velocity yuX 2 //i 2 . 

The point P lies in the plane OQPC ; and since 
C P _C P _OC f _OC 
CP~ LQ -OL-~OC" 
therefore OC . C P = OC . CP, 

and P lies on the rectangular hyperbola PP ; this is the 
geometrical property principally employed by Prof. Sylvester. 

(Solid Geometry, Salmon, 167, 180; Smith, 163, 167.) 



AND THEIR APPLICATIONS. 223 

The angular velocity of the vector C P with respect to the 

revolving plane C P being ~7?"~Mp & follows that, if p, $ 

denote the polar coordinates of a point P on the herpolhode 
described by P on the revolving plane C P , then 

n 2 OC" 2 K 



, _ X 2 \ A-D.B-D.C-D 



equations similar to those required for the herpolhode of P. 

In particular, if we take \ 2 = k 2 , then 00 = 0, and the con- 
focal OP is a cone ; and the plane through rotates with 
constant angular velocity /UL, while the cone, called by Poinsot 
the rolling and slipping cone, rolls on this revolving plane, 
the angular velocity about the line of contact OH being v. 

If we consider the curve described on this revolving plane 
by the point H, the foot of the perpendicular from P on the 
plane, then p, being the polar coordinates of H ( 113), 



dt ~dt ABG~ ~ ~p 

so that the point H describes on the revolving plane an orbit 
as if attracted to ; and, as in 89, we shall find that the 
requisite central force is of the form Ap + Bp s . 

(Pinczon, Comptes Rendus, April, 1887.) 
This is otherwise evident, by noticing that the vector x+iy 
of this curve satisfies Lame s equation ( 204) 

72 
jj^x + iy) = (2$m + &v)(x + iy), 

where 

so that % 

A value of X may be found which makes the herpolhode of 
P a closed curve ; and this closed polhode is an algebraical 
curve, when v is an aliquot part of a period, the correspond 
ing elliptic integrals of the third kind becoming pseudo-elliptic. 

Abel has devoted great attention to the subject of pseudo- 
elliptic integrals ((Euvres, XL), and the algebraical herpolhode 
affords an interesting application of his theorems ( 218). 



ELLIPTIC INTEGRALS IN GENERAL, 

The Addition Theorem for the Third Elliptic Integral. 
A 215. Theorems (9) and (10) of 189 show that, employing 
the function $(11, v) of 201, 



log 



or 



or i ai 2 = o 

o-^ + ^ -h te. 2 )orv a-u^ju^ 
where, expressed by elliptic functions of u v u 2 , and v , 

U ~ V + U 



Also, as in equation (8), 188, 

log (fr(v, u) = log 0(u, v) + i^fy vf tfc ; 
so that 

log 00, O + log^ , u 2 ) 

= log <f>(v, Ul + u 2 ) - { fa + fu. 2 - ^ + u. 2 ) } v + log Q, . . . (3) 
the Addition Theorem for the parameters u v u 2 . 

These theorems have been generalized by Abel for the addi 
tion of any number of amplitudes or parameters in the 
Third Elliptic Integral, and the proof is a simple extension of 
his method, employed in 162 (CEuvres, XXL). 

Denoting by a any arbitrary quantity, equation (7) of 162 
may be written 

1 dx r Ox r 

a x r *JX, r ~~ (a x r )\j/x r 
Now, since Qa is of lower degree in a than \fsa, and 



it follows that, when resolved into partial fractions, 

6a _ y 6x r 

and therefore, writing fx and <px for P and Q respectively, and 
A for the value of X when x = a, 

^, 1 dx r 6 a 9 <f>aSfa faS</>a 
= V^ = (fa) 2 - 



*J a fa + <t>a 
or 2 



AND THEIR APPLICATIONS. 225 

Integrating, with the notation ( 197, 199), 




so that 

<r(v+U r ) 



n- 
< 

is expressible by elliptic functions, ? and #> , of v ; provided that, 
as in (11), 162, 
t m 

= 0, or 2tt- r = 2u r , (6) 

the coefficients in fa and <pa being determined as functions of 
pu r and $ u r by the plexus of equations (4) in 162 ; fa and 
<p a being the same functions of u r . 
Thus the function 



is an elliptic function of v provided that the sum of the values 
u r ofv which make the function vanish is equal to the sum 
of the values u r which make the function infinite ; in other 
words, briefly expressed, provided the sum of the zeroes u is 
equal to the sum of the infinities u . 

In particular, with the u r s all zero, 1,u r = ; and in equation 
(6), 162, we can put 

\fsa = (fa) 2 



so that 2 log <p(u r) v) = log(fa + </>a . *JA ) + constant. 

Thus 11^,.,*), or ^+%)^+Xl_-HO ) ....... (8) 



when u l + u 2 -{-u 3 +...+Uf Jll = 0, ..................... (9) 



is a rational integral function of $>v and $> v, which may be 
written, as in 198, 



v .............. (10) 

G.E.F P 



226 ELLIPTIC INTEGRALS IN GENERAL, 

So also, since ( 201) 



therefore, writing 7" for %*, 

r=ui-l 

2 log (j)( U r> v) = log 0( U, v) + log Q + a constant, ...... (11) 

r=l 

where Q = C f /(p 17 #w). 

In particular, when 7=e a , 0(#, v) = ^/(pv-e a ) ( 202), and 

r n"J<tt r ,t;)=a/V(fw-e tt ) > .................. (12) 

? = ! 

when u 1 + u 2 + u 3 +... + u f ji-i = (ji) a . 

By an interchange of amplitude and parameter, 

2 log (f>(u, v r ) 2 log 0(i&, yV) = log Q pu, .......... (13) 

provided that 2^ = 2^ 

12 being a function of $m, ^ u, $v t $> v ; and 

p = 2(fy r -f y / r). . 

216. A further application of Abel s Theorem of 162 shows 
that p is expressible as a function of pv and $ v\ this is the 
generalization of the Addition Theorem for the Second Elliptic 
Integral, given in 186. 

For 



and this case can be determined as a degenerate case of the 
preceding result ; since, making a oo , 



Ja-X T JX T 

= the coefficient of I/a 2 in the expansion in ascending powers 

1 . fa da. ,JA /-, 1X 

of I/a of -^ log^^j ...................... 0- 

Thus, with Z = 4fl3 3 - # 2 a - ^/ 3 , and a; = py, 
then 

and p 

Jacobi calls V^ the /^c^o? o/ tae 27iird Elliptic Integral 

(Werke, II., p. 494.) 



AND THEIR APPLICATIONS. 227 

217. Similar results hold when, as in 167, X is supposed 
resolved into two factors, X l and X. 

Denoting P^ - Q*X 2 by ^x y 

and varying the arbitrary coefficients in P and Q, and conse 
quently the roots of \^x = 0, as in 162, then 

^x r dx r + 2PSP . X - 2QSQ . X, = 0, 
while PJX^+QJX^Q , 

so that ^ x r dx r - 2(Q3P - PSQ^X^) = 0, 

or far = 2 Q-dP-PSQ^ Ox r 

^/X r \]s x r \ls x r 

and I,dx r /^/X r = 0, or 2u r = 2u r . 

Again, as in 215, 

2_L_ dx r _ Oc<^ _ 
a-x r ~ ~ 



^ loo , 
* 



Thus, as an application to the formulas of 174, 176, 186, 
and 189, take, as in 38 (Durege, Elliptische Functionen, 36), 

X = X 1 X 2 , where X^ = x, JT 2 = (1 #)(! kx). 
Then, with x = sn 2 u, 

c rxdx 2 , 



and 



; 

in Legendre s notation, with = arn i^,, and ??-== I/a. 
Now, if, as in 164, 165, we take 

P or ixp-\-x, and Q or (f>x = q, 
and denote by x v x 2 , x 3 , the roots of the equation (7), 167, 

V*, or P^-Q 2 ^, or 
then 



x l + 
so that, as in 164, 

(2 - ^ - ;r 2 - a; 8 + kx^x^ = 4(1-^.1-^.1- a; 3 ) f 
where 



228 ELLIPTIC INTEGEALS IN GENERAL, 

AgaiD, 

a dx r a ^Xt, , f a 
a-x l ^/X 1 Ja-Xc, J~X 9 Ja-x 3 



or - tanh - *, (16) 
V^ la^/Ai 

since a^, a? 2 , & 3 vanish when p and g are made zero ; and this is 
equivalent to the result of equation (9), 189, with a= 1/n, 
A l-a.I-ka 



and 

fa 



Similarly, for the Second Elliptic Integral, 



00 



= -I* TET f^- C* tanh-"^/-*; 1 --^ (a=x} 



(17) 
as before, in 174, 176, and 186. 

218. Abel s pseudo-elliptic integrals are derived by making 
the u s equal in equations (7), (12) ; or the v s equal in equation 
(13) ; also by making their sum equal to a period a , or the 
sum of multiples of periods, such s^sp^ + qcoy 

Now /JL log 0(it, v) is of the form log 2 pu, 
or <j)(u, vY is of the form e-P M Q, 

where Q is a rational integral function of <@u and <p u of the 
form of in (8), sometimes qualified by a divisor ^/(^u e a ). 

We begin with the simplest case of an algebraical herpolhode 
by taking < y = ft) 1 + Jw 3 ; and then, from equations (39) and (40), 
54, we can infer that the value of s, between e l and e 2 , which 

makes &JZ3?.iV~ e s _ e i _"~ e 2 e z ~ e z 



s 8 or v = 



AND THEIR APPLICATIONS. 229 

Denoting pu by s, p u by ^/S, and pv by a, we infer that 



is pseudo-elliptic, that is, can be expressed in terras of 

/ds/JS and of i&u-\QJS/P). 
In fact, by differentiation of 



_ . 



since ip v = - 2 /v /(e 1 - e 3 . e 2 - e s ) { ^/(e l - e 3 ) - ^/(^ - e s ) }. 
In the herpolhode, therefore, of 113, 



or = - ^ + H s /(e 1 - e s ) - J(e 2 - e.)} nt, 

and therefore, relatively to axes revolving with constant 

angular velocity, 



the herpolhode will be the algebraical curve, given by 



a s 

(a s) cos 20 = ^/(s e l .s &,), 

(a - s) 2 cos 2 2(9 = (a - s) 2 - (e 3 + 2 a )(a - ) + (a - e.Xa - e 2 ), 
(a - s) 2 sin 2 20 + { VK - 6 3 ) + v/(* 2 - e 3 )} 2 ( a - s) 






where, as in 1 1 3, a s, or p u pu = 2 

7I/" 

Referred to Cartesian coordinates, in which 



this equation becomes 



of the form ( 2 + 6 2 )(i/ 2 +6- 2 ) = a* ....................... (18) 



230 ELLIPTIC INTEGRALS IN GENERAL, 

The relation $v e% = ftt /(e 1 e^.e 2 e s ), 

combined with the equations of 110, 113, leads to the relation 

A-D.D-C_A-B.B-C . 
D 2 B 2 

and either B = D, which gives the separating polhode ; or 

D = A~B + C 

the relation for this algebraical herpolhode. 
Now, from 108-110, 

(D D\ 2 /x 2 (D D\ 2 /x 2 

while, with A > B > D > G, and e e = e v e a = e z , e b = e 3 , 



^1 ^2 ^c ?a (Q 

To determine the confocal surface which will describe this 
algebraical herpolhode by rolling on a fixed tangent plane, we 
must equate the angular velocity of the axes to jwA 2 /A 2 ; and 

X 2 1 



The squares of the semi-axes of the confocal are therefore 

_l_ 1 

A 2 2 

D 1 I 

B-2- 
D 1 1D 



while the square of the distance from the centre of the tangent 
plane on which this confocal rolls is given by 



The confocal is therefore a hyperboloid of two sheets, of the 

0,2 ,,,2 ~2 

-S-tftS- 1 

arid in rolling on a fixed tangent plane at a distance b from 
the centre, it will trace out the algebraical herpolhode (18), 
being the preceding herpqlhode, changed in scale in the ratio 
of h to b (Halphen, F. E., II., p. 285). 



AND THEIR APPLICATIONS. 231 

219. A more complicated case can be constructed by taking 
v a?! + Jo) 3 ; but now we must choose particular numerical 
values for # 2 and </ 3 . 

If we select the modular angle of 15, then 2or = J, and in 
(C), 53, J= 5 3 -7-4, J- 1 = H 2 H-4 ; so that, by choosing A = 108, 
then # 2 = 15 #3 = 11 ; 

and i = i-r-V 3 e. 2 =-l, e^ = ^-^3. 

It is easily verified that, with the above value of v, $>v = ^\ 
for p2v= f = p4i>; also this value of <pv or s makes, in equa 
tions (39) and (40), 54, 

Je, e 9 . e 9 ?o \ if e i~~ e z- e i~ e ^ ^ 

* ( J -f^ ; fe -) =2 P (--^; h ; ^ -4 

The corresponding elliptic integral of the third kind in the 
herpolhode will now be pseudo-elliptic ; we find, in fact, that, 



if 0= . sin - 1= 

(2s -1)* 
dO 1 2s + 5 1 



since i$ v= 3^2 ; so that, in the herpolhode, 

flip vdu . A 

<t>-nt=/ - -=-i x /2 ?i^ + 0; 

y pv-fwi- 

and therefore, relatively to axes revolving with constant 
angular velocity /u. ^^/2n, the herpolhode will be the alge 
braic curve 



(2s- 1) 
or (l-2s) 3 sin 2 30 + 9(l- 2s) 2 - 108 = 0, 



in which 1 - 2s = 2(pv - $>u) = 2^ 2 ^ = 3^, suppose ; 

71" /i" C" 



and now ^sin^O + 3c 2 p 4 - 4c 6 = 0, .................. (19) 

a curve, consisting of six equal waves, arranged on a circle. 
With (i.) A > B > D > C, and 



2/4 T) T) 

then (113) 



so that 



A-D.D-G A-D.B-D 



232 ELLIPTIC INTEGRALS IN GENERAL, 

Then, either A D = 0, which would give a stable rotation 
about the axis A ; or 

zr;B+(7 ; .......................... < 20 > 

so that D is the harmonic mean between B and C. 

3 ^B-D.D-G 
Again, pv-e a = ^-z- -^c~ ~> 

so that ~ 






= 

which is impossible, with A > B > C. 

But (ii.), with A > D> B > C, we find that D is the har 
monic mean between A and B] also 



i-.I-x/Sf! n 11 2_V3/1_1\ 
D A~ 2 \D BJ B + C~A~ ~2\C B) 



so that 2 + ^/3 is the ratio of the semi-axes of the focal ellipse 
of the momental ellipsoid, and /3(^/3 l) is the excentricity. 
Another algebraic herpolhode can be constructed by taking 
|co 3 ; and, with <7 2 = 15, <7 3 = 11, we find that 



Now, if 
in . 1 6 



^ y 2(^/3-1) 
ds~ 
so that 



/%p vu_ 
$>v-pu~ 



(28-2^/3 + 



and now the algebraic herpolhode, with respect to revolving 
axes, is given by 

(28-2^3 + 5)^1 30 = 6(^/3- I)*J(s-e 2 . s-e 3 \ 
reducing to an equation of the form 

................... (23) 



AND THEIR APPLICATIONS. 233 

With (i.) A > B > D> C, and 



T , , A-D.D-C -B-D.D-G 

Therefore -^ -^ 

and rejecting the factor DC y 



jD-C .4 2^/3-3 1 1 2,7:i-3/l I 
- = r ~= - 



,, 

-A-D 0= 6 r 




so that the excentricity of the focal ellipse of the moniental 
ellipsoid is ^3 1. 

With (ii.) A > D > B > C, we are led to an impossible result. 

Points of Inflexion on the Herpolhodes. 

220. The original herpolhodes drawn by Poinsot (Theorie 
nouvelle de la rotation des corps) were represented with points 
of inflexion, as curves undulating between two concentric 
circles on the invariable plane. 

But it was pointed out by Hess, in 1880, and de Sparre 
(Comptes Rendus, Nov., 1884), that such points of inflexion can 
not exist on Poinsot s original herpolhodes, which are curves 
always concave to the centre, as drawn in Routh s Rigid 
Dynamics, Chap. IX. ; like the horizontal projection of the path 
of the bob of a conical pendulum, or like the path of the Moon 
relative to the Sun, a good figure of which is given in the 
English Mechanic, p. 337, June, 1891, by Mr. H. P. Slade. 

The herpolhodes described on planes parallel to the invari 
able plane in Sylvester s representation are capable, however, 
of possessing points of inflexion, when the confocal of the 
momental ellipsoid attains a certain shape. (Hess, Das Rollen 
einer Fldche zweiten Grades auf einer invariabeln Ebene* 
Munich, 1880 ; de Sparre, Comptes Rendus, Aug., 1885.) 



234 ELLIPTIC INTEGRALS IN GENERAL, 

Denoting by h the constant distance from the centre of the 
plane upon which a quadric surface rolls, de Sparre shows that 
the herpolhode on the plane has points of inflexion, when the 
quadric is 

(i.) an ellipsoid 

5+S+S- 1 - <<* if W^.o-d^ + i; 

(in a momental ellipsoid, A<B + C, or z < -i + - i > so ^hat 

points of inflexion cannot exist on the herpolhode) ; 
(ii.) a hyperboloid of one sheet 

/g2 y2 z 2 111 

"2 + ?5 ~~ "9 = 1 a 2 <b 2 , if A, 2 < a 2 , and -s > r 2 + ~o 
a 2 6 2 c 2 a 2 b 2 c 2 

(iii.) a hyperboloid of two sheets 

x 2 v 2 z 2 111 

-o T7 , = 1, 1 2 < c 2 , if 70 > -o + -o, whatever the value of /i. 

a 2 fc a c 2 6 2 a 2 c 2 

These herpolhodes being similar to the original herpolhode 
of the momental ellipsoid, when referred to axes rotating with 
constant angular velocity /xA 2 /A. 2 , can be considered as defined 
by the polar coordinates p, 0, given in terms of the time t, by 
the equations of 113, 

p fs5S #(fW fW), (1) 

d9 vifi v /n\ 

= m + - ~n (2) 

at $>v $u 

with u = nt + up v = (0 l + t a) 3 , m/ju = l \ 2 /h 2 . 

Denoting the velocity in the curve by F, and its radius of 
curvature by R, then, resolving normally, 

V*_d P I d( $e\ dO(<]*p 
R~ dt P ~dt\ p dt) p dt\dt 2 

which will be found to reduce to an equation of the form 
F 3 

_ == P / 

wh ere P = m 3 -f- 3mn 2 $)v + nHp v, 

Q = $m*nip v mn 2 $>"v ^nH$> "v ; 

and the corresponding herpolhodes will have points of inflexion 
when X is chosen so that Pp 2 -f Q can vanish. 

Thus Halphen points out that the algebraical herpolhode 
of 218 will have points of inflexion, if b 2 < Ja 2 . 



AND THEIR APPLICATIONS. 235 

221. The polhode being given by the intersection of the two 
quadric surfaces Ax 2 +% 2 + Cz* =M 2 , 



we may in consequence write 




c 

where (BC)c 

A(B-C)a* + B(C-A)b 2 

and then 



2 +X 6 2 +X c 2 +X 
the equation of a system of confocal quadrics, on choosing I 

B-CC-AA-B 
such that I = . 1 D I TT~ 

Then 

n n ^A]^ 



2 
a ~ b - 



By varying X along the polhode, we find 

2 dx 1 d\ dx \ x d\ 

_ __ ______ _ __ OT" - _ _ _ 

xdt7a*+\dt ~dt 2o*+X(ft 

so that the polhode is an orthogonal trajectory of the confocal 
surfaces, for any one of which X is constant ; and two ellipsoids 
can be drawn on which the curve is a polhode, of which the 
generating lines of the confocal hyperboloid through the points 
are normals. 

When these confocals are hyperboloids of one sheet, the 
generating lines may be made of material rods or wires, 
jointed at the points of crossing ; and now any such a system 
of rods forming a hyperboloid is capable of deformation, and 
assumes in succession the shape of the confocal hyperboloids ; 
the trajectory of any fixed point on a rod being orthogonal to 
the hyperboloids, and therefore capable of being a polhode, if 
the hyperboloids are coaxial with the momental ellipsoid of 
the body. (Messenger of Mathematics, 1878 ; Senate House 
Solutions for 1878 ; Larmor, Proceedings Cam. Phil. Society, 
1884, Jointed Wickerwork] Darboux and Mannheim, 
Rendus, 1885 and 1886.) 



236 ELLIPTIC INTEGRALS IN GENERAL, 

Darboux has shown (Despeyrous, Cours de mecanique, t. II., 
Notes XVII., XVIII.) that if we hold a given generator fixed, 
then any point fixed in any other generator will describe a 
sphere ; thus, if a rod moves with three points P, Q, R on it 
connected by means of bars to three fixed centres A, B, G in 
a straight line, any other point 8 of the rod will describe a 
sphere about a centre D in the line ABC, such that the A. R. 
(ABCD) is equal to the A. R (PQRS). 

The point where the line PQR meets the generator parallel 
to ABC will describe a plane, the corresponding centre being 
at an infinite distance ; and generally, if one generator is held 
fixed, any point on the parallel generator will describe a plane. 

The herpolhode can now be described by taking a jointed 
hyperboloid, similar and similarly situated, and of half the size 
of the former one used for describing the polhode, with one 
generator fixed along the invariable line 0(7, and with the par 
allel generator along the normal PQ at P ; and now, if P is 
moved in a direction perpendicular to the hyperboloid at P, 
it will describe a plane curve, which is the herpolhode. 

222. Any point fixed in a body moving under no forces, 
whose co-ordinates with respect to the principal axes are 
represented by a, b, c, will have component velocities 

cq br, ar cp, bp aq, parallel to the principal axes; 

and will describe a curve whose projection on the invariable 
plane will be given, in polar co-ordinates p and 0, by ( 104-113) 



(bCr- cBqf + (cAp - aCr) 2 + (aBq - bA cff 
~~~ 



+ {(a 2 + b 2 )r - cap - bcq }~ > 

the moment of the velocity about the invariable line OC ; and 
p, q, r are given as functions of t in 32, IOC, and 108. 



AND THEIR APPLICATIONS. 237 

The equations are much simplified when the point is fixed on 
one of the principal axes, when two of the three quantities 
a, b, c vanish ; and it will be a useful exercise for the student 
to prove that, in these cases, the curve of projection on the 
invariable plane with respect to axes rotating with angular 
velocity G/A, G/B, G/C respectively, is given by an equation 
of the form 

x + iy = k<f>(u,(a a v), or k<f>(u, w 6 -f), or k(/>(u, (o c -v). 

Another useful exercise is to deduce -Poinsot s relations when 
the co-ordinate axes fixed in the body are not principal axes. 

Now, if the equation of the momental ellipsoid is 

Ax 2 + By* + Cz 2 - ZA yz - 2B zx - 2C xy = Dh 2 ; 
and if p, q, r denote as before the component angular velocities, 
and h lf h. 2 , h s the components of angular momentum about the 
axes, the three equations of motion under no forces are 



where 

h^Ap-C q-B r, h, = Bq-A r-C p, h 3 = Cr-B p-A q; 
and these equations are solvable by elliptic functions. 

(Dissertation Ueber die Integration eines Differentialgleich- 
ung ssy stems ; Paul Hoyer, Berlin, 1879.) 

223. The numerical results obtained in the preceding alge 
braical herpolhodes can be utilized in the corresponding 
problems of the revolving chain ( 205-206) and of the 
Tortuous Elastica ( 207). 

Putting t = J, or v = o> 3 in 206, 
then p-y = e 3 - J(e^ - e 3 . e. 2 - ej, 

. 2 - ej} 



pu-pv 

^^^^ 

or (s-^)cos[2^+ { V(i-*s) + V(2-*3)}W] = ^/(s-e^s-e^ 
where s-pv = r*/k\ 

In the corresponding problem of the Tortuous Elastica of 
207, it is merely requisite to replace x by the arc s. 



238 ELLIPTIC INTEGRALS IN GENERAL, 

The working out of the analogies for the other algebraical 
herpolhodes is left as an exercise; merely mentioning that 

jKK; 15, H)= -f, 
and that, if 



= J sin- = j cos- 

(2s + 3)* (2 

dO 1 2s + l 1 ^ * 






\i<& vdu u I . , W 

= -75 - , sm - * ^- 

$m-py ^2 3 (^ 



224. The analytical expressions in 208, 210 for the motion 
of the Spherical Pendulum and of the Top or Gyrostat show, 
by comparison with the equations of the herpolhode in 200, 
that this motion may be considered as compounded of two 
Poinsot representations of the motion of a body under no forces, 
as given in 104, 214 (Jacobi, Werke, II., p. 477). 

The relations connecting these two component Poinsot 
motions have engaged the attention of Darboux (Despeyrous, 
Cours de me canique, II., Note XIX.), of Halphen (F. E., II., 
Chap. Ill), and of Routh (Q. J. M., XXIII.). 

We may put the conclusions arrived at by these mathema 
ticians in the following condensed form, depending on funda 
mental dynamical and geometrical considerations. 

(i.) If the vector OH represents the axis of resultant angular 
momentum, then H lies in a horizontal plane through the point 
G, where the vertical vector OG represents G, the constant 
component of angular momentum about the vertical. 

(ii.) If the plane drawn through H, perpendicular to the axis 
of the Top, cuts this axis in G, then 00= Or, the constant com 
ponent of angular momentum about 00, the axis of the Top. 

(iii.) These two planes, one horizontal and through G, which 
we shall call the invariable plane of G, and the other through 
and perpendicular to OG, which we shall call the invariable 
plane of G, intersect in a line HK perpendicular to the vertical 
plane GOG and if HK meets the plane GOO in K, then 
CR 2 - GH 2 = OK 2 - GK 2 = OG 2 - OG 2 = G 2 - 2 r 2 . 

(iv.) The instantaneous axis of rotation 01 lies in the plane 
HOC ; and if 01 meets GH in I, the resultant angular velocity 



AND THEIR APPLICATIONS. 239 

about 01 is OI/C; also CI/CH=C/A, 
and the velocity of C is r . CI. 

(v.) By equation (i.) of 210, the square of the velocity of G 
is (:>C- 2 r 2 Wg/A)(c-hcos6); 

so that CI 2 = (2C 2 Wg/A)(c - h cos 0), 



= 2 A Wghk(pw $>u), suppose. 
Then, by equation (3) of 210, with u = nt + u> 

%ln-k*p" 2 u = gtf(pu - pa)(pu - &b)(pu - pw) - (a 
and therefore, when u = a, b, w, we have three equations of the 
form i$> a = a + /3pa, i$> b = a + /3$>b, if r w 
so that, according to 165, we may put iv = b a. 

(vi.) Now GH 2 = 2AWghk{&(b-a)-&i<,} 

2 A Wghk(f>w -fru,), suppose, 
where pi</ - p(a + b) = - (G 2 - C 2 r 2 )/ 2A Wghk ; 

and since 

. G-Cr 



and 

therefore ptg - f(b - a) = - (p 

and therefore ( 151) we may put w = 

(vii.) The point JT moves in the invariable plane of G with 
velocity equal to the impressed couple of gravity, and parallel 
to the axis of the couple ; so that the velocity of H is in the 
direction HK, and equal to Wgh sin ; and the moment of this 
velocity about G is Wgh sin 6 . GK. 

But GK sin 6 = OC-OG cos 0, 

so that p 2 (d<j>/dt) = Wgh(Cr - G cos 6), 

if p, denote the polar coordinates of H in the invariable 
plane of G. 

Now p z = 2 

and cos = 

so that finally we shall find, after reduction, 



__ _ 
dt 24 



and therefore H describes in the invariable plane of G a her- 
polhode with parameter 6 + a. 



240 ELLIPTIC INTEGRALS IN GENERAL, 

(viii.) Similar considerations will show that the curve de 
scribed by H in the invariable plane of C is also a herpolhode, 
with parameter b a. 

If in equation (2) of 210 we replace Or by AT, the motion 
of OC is unaltered, but now the momental ellipsoid at becomes 
a sphere, and OH is the instantaneous axis of rotation ; so that 
the motion of OC is produced by rolling the cone, whose base 
is the herpolhode described by H in the invariable plane of 0, 
on the cone whose base is the herpolhode in the invariable 
plane of G, the angular velocity being proportional to OH. 

(ix.) But in the general case, where 01 is the instantaneous 
axis, the curve described by / in the invariable plane of G is 
similar to the curve described by H, and is therefore a herpol 
hode. 

Now from (v.), drawing CM, IN perpendicular to OG, 



20 2 Wg/A)(c- 

A -s . GN 



so that O/ 2 varies as the height of /above a certain horizontal 
plane ; and the locus of 1 is therefore a sphere, to which the 
point and this plane are related as limiting point and radical 
plane. 

The motion of the Top can therefore be produced by rolling 
the herpolhode described by / in the invariable plane of G on 
this sphere, with angular velocity proportional to 01. 

(x.) It still remains to be shown that the cone described by 
01 in space round OG is a herpolhode cone ; this is left as an 
exercise. 

Darboux shows that two such hyperboloids as those described 
in 221, with a pair of generating lines, PQ, PQ in coincidence, 
and the opposite generators OG, OC of the same system inter 
secting in a fixed point 0, may be used to represent the 
motion of OC, the axis of a Top, when OG is held vertical; 
the point P of intersection of the coincident generators being 
made to describe herpolhodes in the invariable planes of G 
and G, by being moved in the direction of the common normal 
of the hyperboloids. 



AND THEIR APPLICATIONS. 241 

225. The numerical results of the pseudo-elliptic integrals 
of 218, 219, and 223 can be utilised for the construction oi 
similar degenerate cases of the motion of the Top. 

Thus, if a = I o> 3 , b = ^ + io? 3 , 

then 6 + a = o) 1 + o) 3 , 6 c^ee^; 

and we shall find cos a = 0, cos ft = K, cZ = sec /3, and 
C V = 2 A Wgh sec ft, G 2 = 2 A Wgh cos ft. 
The spherical curve described by C is now given by 
sin sin(nt cos ft \ts) = ^{cos 0(cos ft cos 6)}, 
sin cos(nt cos ft \fs) = ^/(l cos ft cos #). 
With a = Jo> 3 , 6 = M! |o> 3 , and 6 + a = o) v 

we find that cos a, cos ft, and d are unaltered, but Cr and G 
are interchanged ; and C now describes the spherical curve 

sin sin(nt \[r) = *J{cos #(sec/3 cos 0)}, 
sin cos(nt ^) = ^/(l sec ft cos 0). 
Again, with a = f o? 3 , 6 = ^ J o> 3 , (/ 2 = 1 5, # 3 = 1 1 : 
so that ^a= f, p6 = J, we find that 



and the spherical curve described by G is given by 
sin 3 $ sin 3\fs = ( 1 2 cos 0)*, 
sin 3 0cos 3x^ = (l + cos ^ + cos 2 0) v /(2-f 2 cos# cos 2 0). 

To realise this motion practically, place a homogeneous sphere, 
of radius c, inside a fixed spherical bowl of radius a, in contact 
at an angular distance of (50 from the lowest point, and spin 
the sphere about the common normal with angular velocity 



The sphere if released will roll on the interior in this curve 
As another numerical illustration we may take 



when 



Also, with 9-2 = 30, 

p-J3 = - 5 - f /6, 

G.E.F. Q 



242 ELLIPTIC INTEGRALS IN GENERAL, 

226. It is convenient to represent the two parts of \/s by 
^ and i/r 2 , such that 



"va-i G + Cr l 
dt 2 A 1+C080 du 

1 G-Cr 1 



dt 2 A 1 cos du <pb pu 
also to put x = V r~V y 2 whence Euler s an^le <p = 
and dx = Cr-Gcos9 ( 

cw A sin 2 
an expression obtained by interchanging (7 and O in i/r. 

With ct = pw 3 , 6 = o) 1 + gco 3 , a change of g into q interchanges 
(} and Cr, while a change of p into > interchanges G and 

O: both changes of sign change G and G and 6V into 

Cr, and thus reverse the motion. 

The following degenerate cases of the motion of the Top will 
afford an exercise on the preceding results of 210, 224 : 

A. With b-a = (a lt or q p = Q, 

^_ G _ c _ 1 + cos q cos /3 
Cr k cos a + cos /3 
C 2 r 2 / 2A Wgh = cos a -f cos ft ; 
and by 215, x is now pseudo-elliptic ; and 

X ~ \/( cos a ~h cos /3)x/(i#/^X ~~ 
i , , . / (cos 8 cos 0)(cos cos a) 

^J^gJ g /7__foy^-l / \ I / \ / 

V I + cos a cos /3 (cos a + cos /3)cos 

i>s/{( cos /3 cos 0)(cos cos a)} 

= sin " 1 ^- ^ . \ 

sin t7 

_ 1>v /{l + cosa cos /3 ( cos a + cos /3)cos 0|- 

The angular velocity of H round G in the invariable plane 
or G is now constant and equal to $G/A. 

B. With b a = o^ + w 3 , or q p = 1, 

n_ G _c _I+d cos a 
Cr~Ji~ cos a + cZ 



and the spherical curve described by C has cusps on the circle 
given by 0=8 , and now 



-, , / (d cos0)(cos0 cos a) 

where =tan~ \ / - T 7X ^ etc. 

V 1 + d cos a - (cos a + a)coa 



AND THEIR APPLICATIONS. 243 

The angular velocity of H round G is again equal to %G/A. 
C. With b + a = u v or q + p = 0, 

7 Or 1 -f cos a cos 8. 
a = -77- = - - - TT > 

G cos a + cos p 

and now \Js is pseudo- elliptic, and given by 



while the angular velocity of H round C in the invariable 
plane of C is constant and equal to ^ 
D. With /_> + a = a?! + o> 3 , or g+p = l, 
(7 r 1 + r? cos a 



-^= j 
Cr cos a + a 



and the angular velocity of # round C in the invariable plane 
of C is again %Cr/A. 

E. With g = 1, 6 = o^ + 3 , G Cr = 0, and i/^ 2 disappears ; and 
now cosj3 = c/A=l, the Top being spun originally in the 
upright position. 

Now if the Top falls ultimately to the extreme inclination a, 
we find that C 2 r 2 /2A Wgh = 1 + cos a ; 

and subsequently, after a time t, 

sin ^0 = sin Ja sech[sin \a^/(gjl)t} t 
Crt /cos cos a . 

^2^- 8ln V i 



so that the integrals for t and i//- are pseudo-elliptic. 

F. With q = Q ) b = w l ,G Cr = 0, and i/r 2 again disappears; but 
now cZ = l, and the Top does not rise to the vertical position. 

For numerical illustrations of this motion, take 

a = fo> 3 , and </ 2 =lo, # 3 = 11, when ^a=f; 
or ^r. 2 = 48, ^r 3 = 44, when pa = 4. 

G. With p = l, a = co 3 , (r + (7r = 0, and ^ disappears; now 
cos a = 1, and the Top passes through its lowest position. 

For numerical examples of pseudo-elliptic cases, employ the 
results pK + ift^; 15, 11) = 1, and ^(^ + -^3; 48, 44) = 2. 

H. Withp=l and q=I, G = and Cr = ; and the motion 
reduces to plane revolutions, as in 18. 

I. With p = l and q = 0, G = and (7r = 0; and the motion 
reduces to plane oscillations, as in 3. 

K. With > = !,# =0, d=l, cos/3= 1, cosa= 1, the pen 
dulum is at rest in its lowest position. 



244 ELLIPTIC INTEGRALS IN GENERAL, 

The Trajectory of a Projectile, for the Cubic Laiv of Re 
sistance. 

227. An immediate application of the function 0(u, v) of 
201 occurs in the solution of the motion of a body under 
gravity in a resisting medium, in which it is assumed that the 
resistance of the medium is in the direction opposite to motion, 
and that it varies as the cube of the velocity. 

Refer the motion to oblique coordinate axes, one Ox in the 
direction of projection at the point of infinite velocity, and the 
other Oy drawn vertically downwards. 

Denote by w the terminal velocity of the projectile in 
the medium ; so that if W denotes the weight in pounds, the 
resistance of the air at a velocity v is a force of W(vjw) z 
pounds, and the retardation produced is g(v/w) B . 

The equations of motion are then 

dx 



= g(ds\ 
~ ^AdtJ 



ds " " ( ) 

g ds\*dy 

+J " 



Eliminating the term due to the resistance, 
dx d 2 y d 2 x dy _ dx 
dt dt-~"dP dt~~ 9 !K 

or, writing p for dy/dx, 

dp dt dp dx 

- 



01- 



It Ox makes an angle a with the horizon, then 
ds 2 _dy* 2 ^^sin</ 

d&^ dt* 2 di dt** 



and now equation (1) becomes 
dfa _ 9( 
dt 2 ~ uAdt dt 



AND THEIR APPLICATIONS. 245 

Integrating, noticing that dxfdt = oc , when p = 0, 



- 3 



suppose, where _p 3 3^ 2 sin a + 3p is denoted by P; 
dx _* 
2T wP - ^ 

Then, from (3), d f- = q(~\~ = ^ 9 P > 

dx *\dt/ ur 



so that 4^=M 



w 2 
and ^=/p-3 / 7, 1 (6 \ 



; (7) 



while ^ = p, 

di w 

n+ f* i 

(8) 







228. The integration required in (6) is similar to that of 
ex. 8, p. 65, discussed also in 157; we substitute 



where m is some arbitrary constant factor ; and then 

40 s - gr s = { (4m 6 - g 3 )p* - 1 2m*p sin a + 1 2m 6 }/j9 2 , 
which is a perfect square, when 

4m 6 # 3 = 3m 6 sin 2 a , or </ 3 = m 6 (4 3 sinV) ; 
so that J(z* -flrg) = mV3(2 -p sin a )/p, 

and 






dp c/dx 



on choosing m 2 = J ; so that 

gx C* & 



246 ELLIPTIC INTEGRALS IN GENERAL, 



Then 



and supposing x = a at the vertical asymptote, where p= ao , 

na since ga I 
^ = 3 V* = 3 



so that 



,</a /##_ 2 
V~ F i<7 2 ~3p 

vg 



cfy 3 w z / lft v 

or x>= /= = -; (W) 



dx ,qa ,gx ,ga ,gx" 
@ ~ - <@ y - ~ 

o ^/.-2 n,,2 5 /,2 



,, 9 <7tt 
6^ 2 r, 
5 W* 



and, integrating, y=f- dx, 







the equation of the trajectory. 

It is convenient to write w and v for gx/w 2 and ga/w 2 ; 



and now 



to be integrated by the preceding rules of 198. 
Rationalizing the denominator np r v $ u, it becomes 

since <7 2 = ; and resolved into linear factors, it becomes 



where CD, o> 2 denote the imaginary cube roots of unity, viz., 

= - \ i + ix/ 3 ^ " 2 = - J 
Now, resolved into partial fractions, 



| . 

2 > v 9 u 2 cov ^ 2 



2 

"T~ 2-i^ "T" "~W 

pu 2 $ 



on making use of the results of 196, when g 2 = Q- 
Then 



g y ri PVH/^ ri &v& du+af fift<*>+f d 

w 2 J 2$v $)u j 2$>(jov $)u J 2$) u?v <pu 
which is prepared for integration as required in 198; and since 



AND THEIR APPLICATIONS. 



2 pv 

= log o-(v u)-\- log o-u uv + constant 



therefore the result of the integration may be expressed by 



The conditions of Homogeneity of 196 also show that the 
last equation (13) may be written 



(TV (TO)! 

or simply 

^ = - 3uv - log ar(v -u)-u> log <r((jov - u) - or 

subject to the condition that y = 0, when u or x = 0. 

The equation is left in the complex imaginary form, as there 
exists no theorem for the expression of 

]ogcr(ft>v u) in the form P + iQ; 

unless we introduce a new function <!>(, a), defined by 
(Halphen, F. E. t I., p. 151) 

/"a 



229. For the expression of the time t in the trajectory, 
equation (8) leads to 



w J pv to-u 



/I p v+p u 7 r\ <p (0v+$> u, r\ <Q ufa+tiu-j 
7;- du+ur/ s ~ -dii+co/ - --du, (lo) 

2 tov - to\i J 2 wv -$u J 1 pw*t> - p n 

when resolved, as before for y, into partial fractions ; so that 

at 

y - = - log 0( - u, t<) - o) 2 log 0( - , o>r) - o) log 0( - 16, w 2 v), 



01 , , 

co-log w log ^ s 

cr % era)? era)- 

or simply 

= log a-(v u) o 2 log o-(c*)V u) co log <r(o) 2 y u), (1 6) 
subject to the condition that = (), when x or = 0. 



248 ELLIPTIC INTEGRALS IN GENERAL, 

By addition, 



I i 



- 

{ crV crU j (TV (TCOV (7(JD 2 V (T 3 U 

and this last term, when expressed in a real form, is equal to 



(Halphen, F. K, I., p. 232.) 
This can be proved independently ; for 



J ~Sv> u 



2 $ v <@ u 

constant 



230. For the purpose of the expression of y and t in ascend 
ing powers of x or u t it is useful to employ the function 

e v ^ v , which we may denote by i/>-( 1(,, t?) or \fs ; 

(TV 

so that \j/( u, v) = a-u (j)( u, v), and i/r = l, when u = (). 

We may now write 
gy/w 2 = log i/r( u, v) w log i/r( w, w y) o> 2 log \[s( u, w 2 v), 

gt/W = log \/s( U,V) ft) 2 log l/r( &, ft)V) ft) log \/r( ^, ft) 2 l ). 

Differentiating logarithmically, 



on expanding the second side by Taylor s Theorem ; so that, 
integrating again, 

77 2 /i* 3 77 

log^(~u, ?;)=- 2 y^ + ^-^^+..., ............... (18) 

Then, with </ 2 = 0, and ^a)V = o)^v l etc., 

/ 2 V/ 

/ y-ft)Vi;+..., .......... (19) 



(20) 



AND THEIR APPLICATIONS. 249 

so that 2 = 3 ^V->i- + ,_ ... (2 1) 



Oi 

and here u = gx/ iv 2 , g. 2 = 0, g B = T T T (4 3 sin 2 a), p; = I , 



231. When p v p 2 , p 3 denote the values of p corresponding to 
three points defined by the values x v x. 2 , X B of x, or u v u 2 , u z 
of u, such that 

X 1 +x 2 + x 3 = 0, or i6 1 -f it 2 + U B = 0, 
then, according to 145, 

This Theorem follows also as a corollary of Abel s Theorem, 
as applied in 166 ; and it is interesting to proceed to the 
determination, in a similar manner, of the corresponding values 
of 2/1 + 2/2 + 2/3 an d *i + ^ + ^ 3 . 

Changing, in 166, x into p and y into P*, then from (7) 166, 

n 

dy 2 + dy 3 )=p l P l - 
3 



z2 



" 



Therefore 



= -log(a-l)-colog(a-w)-a) 2 log(a-co 2 ),...(24) 



= log(a 1 ) - ft) 2 log(a CD) a> log(a co 2 ) ; . . (25) 

P^_Ps Ps_Pi Ps_Pi 

where a = - 2 - --=> - JL=J - f_?_ ; ..... (2 6) 

^2-^3 Ps -Pi Pi~P 2 
and a = ao , when ^ =^> 2 =_p s = 0. 

As a corollary from the preceding expressions for y and t in 
terms of x or u, it follows that 

<r(v u-j)(r(v u 2 )a-(v U B ) _ 1 



o- 3 v a-u l <rU^rU B 



250 ELLIPTIC INTEGRALS IN GENERAL, 

232. By taking x s = and p a = 0, then 

Pi+PtPiPfina=*O t or l/p 1 + l/p 2 = sin 
when a3 1 + a; 2 = 0, or 

Now, from equations (13) and (16), 



= - log =- J S tan - 

2 $> 6 u $rv 



- 0) 



= - log + ^3 tan - 

2 jp% ^ 3 v) v 

In particular, when ^^ = a) 2 , then 



and .^ = _ S 



2*= _ 1 log fc^V V3 tan -- 

^ 4 fy ^2 V 



so that the expressions for y and t are pseudo-elliptic ; and, at 
this point, p = 2sina. 

233. We may now investigate the properties of certain points. 
on the, trajectory. 

When u = 2o> 2 v, 

then $>u = J, <gfu = J sin a, and p = cosec a, 

so that the tangent is perpendicular to Ox. 
The velocity in the trajectory is given by 

iv(p* - 2p sin a + !)*( p 3 - 3^> 2 sin a + 3|?)~*, 
and this is a minimum, by logarithmic differentiation, when 

j9 sing p 2 2p sing + I_ _ 

2 2 si n g -f 1 5 3 ~ 



or p 2 cos 2 a +^ sin g- 1 = ................... (27) 

If the tangent AB makes an angle ft with Ox at the point A, 

^ sin 8 

then p=. f , 

C0s(g p) 

so that the relation becomes 

tang=-2cot2/3 = tan/3-cot/3 .......... (28) 

Then ^(4 + tan 2 g) = tan /3 + cot /3 = 2 cosec 2/3, 
or (3#) = JC 4 ~ 3 sin 2 g) - cos g cosec 28. 



AND THEIR APPLICATIONS. 251 

The relation (28) is equivalent to a number of other re 
lations, such as 

tan(2/3 a) = tan a tan 2/3 = tan a + 2 cot a, 



tan a ={cot(a-)}*-{ten(a -)}*, etc. 

Also, since p = -: r~r~* 

sin a o#> u 

therefore, at these points of minimum velocity, 

^ / 2 w , = T.(4_3sm 2 a) = 3// 3 , and p 3 w=r/ 3 , 
and therefore ?2?<, = #w, or u = fo> 2 , as in 160. 

The integrals for y and t at these points of minimum velocity 
are therefore pseudo-elliptic, and depend on 
f ds , f sds 

J (**- iM 4 * 8 - 1) a ./ (s 8 - we** 8 - iy 

integrals first considered by Euler (Legendre, F. E., I., Cha|>. 
XXVI). 

We find, by differentiation, that 

(29) 

" 



, ...(30) 
^ts- i; tj ^-tr L) -f x/ o V ^" i ; 

-^tan- 1 



x/(4* 3 -l) 

(2 



by means of which the results can be constructed : and 
noticing that, if s = <pv, *J(4-s 3 1 ) = p v, <j<, = 0, g z = 1. then 



we find finally, when u = |cD 2 , 

n-Vly-f^), (32) 



252 



ELLIPTIC INTEGRALS IN GENERAL, 



234. Denoting by the angle which the tangent at any 
point makes with Ox, the tangent at 0, the point of infinite 
velocity, and by the angle which it makes with the tangent 
at A, the point of minimum velocity, then = /3 ( j) 1 and 

sin sin(/3 <) 

L) ~ cos( a - 0) ~~ cos(a 
cosq 



so that 



and 



and since 



sn 



~ 0) - 2 cos ( - ft + 0) 



tan siD - 



c< s(/3 - 0) - cot 25 sin(/3 - A) 
- - - -- 



= - 2 cos a 



_ _9 



= z cos a cosec zp -r-^ - 
sm 



~ 3 sin 2 a) = | cos a cosec 



,, r , 

therefore . ^~^= \-^ -, 






= -^... ...(34) 

tan p <pu-\-$ co 2 

Therefore, at points defined by u v u 2 , where the tangents 
make equal angles with the tangent at A, 

p tt 1 *P /t i ss f *i< k >r 

Thus, if 1^ = 0, then u^ = w. 2 \ and the tangent where u = u>. 2 
makes an angle 2/3 with Ox. 

By the principle of Homogeneity of 196, we can select any 
arbitrary value of </ 3 , and it is convenient to take # 3 =1; and 



gx u , 
l/ -^ = , then 
K; 2 m 



.,. 
now, if 

where m 6 =g 3 , 

With ,(/ 2 = 0, # 3 = 1, we have found, in 166, 



. v (///. ,, 
Again, n - =^, then 
m w 2 

<pv = (4 - 3 sm 2 a)-*, ^ = ^/3 sin a(4 - 
so that, as a increases from to JTT, 
and v increases from w. 2 to |o) 2 . 



cos 
v increases from to 



AND THEIR APPLICATIONS. 253 

Denoting the analytical expression for tan^/tan/3 in (34) 
by X, then X is independent of a or /3, and therefore a Table 
of numerical values of X, with u or mgxju^ for argument, will 
serve for all trajectories. 

It will be a useful numerical exercise for the student to 
prove that corresponding values of u and X are 



2J/-2 
, 1 

21 



0; 



i2 

|0) 2 , 00 . 

EXAMPLES. 
Prove that, with g. 2 = Q, # 3 = 1, 



2. ^ (it - f ft? 9 ) = 

3. u- 



/* ^-^ / > j. \ i ,2cm 1 

. / _ = f u f .73 tanh - */3 
ypit-1 9V p % 

^ " T ^" log p ^ w " ^2) iW 3 tan 
" A log F(U ~ *^) + A v/3 tan 



7. Integrate 



CHAPTER VIII. 

THE DOUBLE PERIODICITY OF THE ELLIPTIC 
FUNCTIONS. 

235. Besides pointing out the advantage of the direct Ellip 
tic Functions obtained by the inversion of the Elliptic Integrals 
( 5), Abel made an equally important step (Crelle, II., 1827) 
in showing that the Elliptic Functions are doubly -periodic 
functions, having a real period, 4>K or 2K, as already defined 
in 11, and an imaginary period, 4>K i or ZK i, where, as 
before in 11, 

o 

Doubly-periodic functions make their appearance when we 
consider functions of a complex argument w = u + vi. 

Denoting x + yi by z y we have already discussed in 179 the 
system of confocal conies given by 

rj = c sin w, or c cos w, when u or v is constant. 

T ,,. r dz 

In this case w= I , 2 2 -, 

J v ( c ^ ) 

and the poles of this integral, as defined in 54, are given by 
z = c, the foci of the confocal system of conies. 
Changing the origin to a focus, then 

r dz 

iv =1 j- 

and z = 2c sin 2 \w, 

2c z = 2c cos 2 Jt0, 
dz/dw = c sin w. 

Denoting by r, r the focal distances of a point, then 
r 2 = (x 4- yi)(x yi) = 4c 2 sin 2 J(u 4- m)sin 2 J(i, w), 

254 



DOUBLE PERIODICITY OF ELLIFIIC FUNCTIONS. 255 



or r= 2c sin 

? = 2c cos 
so that / + r = 2c cos vi = 2c cosh i;, 

/ r= 2c cosu, 

giving the contbcal ellipses and hyperbolas, for which v and u 
are constants. 

It is convenient to denote x yi by z and u vi by i(/ ; 
and now the Jacobian 

T 3(fl5, ?/) 9 . . , 

J or -) = c 2 sm w sm ^o = Jrf . 
9(tt, v) 

236. Now, if we consider the integral (11) of 38, 



then z = sn 2 ^ 

\z cn 



dz/diu = sn \w en \w dn |i(; ; 
and the poles of the integral are given by z = Q, 1, and 1/&. 

Denoting by r, r , r" the distances of a point from these 
poles or foci 0, (7, 0" in fig. 26, then 

r = sn Jitf sn Jt(/, r = en Jw en 10 , &r" = dn \w dn |t(/ ; 
or by means of formulas (2), (3), (5), (28), (29) of 137, with \w 
and Ji{/for it and t , and therefore u and i?j for u + v and it v, 
cnri cnu 1 dn-yi dn u 



r = 

K- 



_ u 4- en u dn t i /c /2 dnvi dnu 

/cr" = ^^^ 



en vi dn 16 en u dn ri 
cnu 



cnvidntt cnuckivi 

From these relations, by the alternate elimination of u and r, 
r + r dn m = en vi\ 
r rdnu =wu ) 

or kr" + ^7^ cn i?i = dn vi"\ 

kr" kr cn u = dn u ) 
or ki-"du vi kr en vi = 1 k\ 

kr"&i\ u kr cnu =1 k) 

the vectorial equations of one and the same system of confocal 
orthogonal Cartesian Ovals (fig. 26) ; also J=krrr". (Darboux, 
Annales scientifiques de Vecole normale supe rieure, IV., 1867.) 



256 



THE DOUBLE PERIODICITY 



As we travel round one of these curves and make complete 
circuits, each enclosing a pair of poles of the integral w, defined 
either by and 1, or 1 and l/k, the integral increases by 
constant quantities 4K" or 4<K i, the corresponding periods of 
the elliptic function sn 2 Jt(;, as in 55. 

y 




Fig. 26. 

By making k = 0, we obtain the degenerate case of the 
confocal conies, and now K=^TT, while K =x> , so that the 
circular functions have a real period 2-rr and an infinite 
imaginary period; on the other hand, the hyperbolic functions, 
as illustrated by the confocal ellipses, have an infinite real 
period and an imaginary period 2?ri. 

Mr. J. Hammond has shown, in the American Journal of 
Mathematics, vol. I., how these Cartesian Ovals may be de 
scribed mechanically, by means of reels of thread, as in the 
case of the confocal conies of 173. 

He takes two reels of thread, of different diameters, fastened 
together, and pivoted on the same axis at C. Now, if the 
threads are led through a pair of the foci, and , the curves 

r IT = c 
will be described, if the diameters are in the ratio of I to 1. 

By leading the threads round an oval, as in fig. 26, theorems 
can be obtained, connecting arcs of confocal Cartesian Ovals, 
analogous to those of Graves and Chasles for elliptic arcs. 



OF THE ELLIPTIC FUNCTIONS. 257 

237. By inversion of this system of confocal Cartesian Ovals, 
we shall obtain another system of orthogonal quartic curves, 
with four coney clic foci A, B, C, D, defined by the vectors 
z = a, ft, y, S, suppose ; and now 

w=fdz/J(za . z-/3 . z- 7 . z-S) ; 
or, writing iv for iv/^/(a y fi~ <>)> then, from 66, 

6 S.z a 91 a 8.z 8 01 a 8.z y , 91 

- 7) = sn 2 ^(;, ? 75 ==cn 2 iic?, - J = dn 2 to. 

a-S.z-/3 2 a-<5.0-/3 a-y.z-/3 

Denoting by r v r 2 , r 3 , r 4 the distances of a point from the 
foci A, B, C, D, then, from these equations, 

Q $ rp R T 

mod. " . -i = sn ^iv sn Aiv , mod. - = en to en Aw 7 , 

a o T 2 a o r z 

mod. - = dn &w dn Ati; ; 
a-y r 2 

so that we obtain the vectorial equations of these orthogonal 
quartic curves on replacing r t r, r" in the equations of the 
Cartesian Ovals by these expressions. 

(Proc. Cam. Phil. Society, vol. IV. ; Holzmuller, Einfuhrung 
in die Theorie der isogonalen Vei^wandtschaften, 1882.) 

238. We now proceed to express the elliptic functions of the 
imaginary argument vi by functions of a real argument v. 

We know that cos vi = cosh v, sin vi = i sinh v, tan vi = i tanh v ; 
and that the function <p or amh u, and its inverse function 
u or amh~ 1 = log(sec ^> + tan 0)= cosh" ^ec^, etc., 
connects the circular functions of <f>, for which /c = 0, with the 
hyperbolic functions of u in 16, for which K = 1 ; and then 
cosh u = sec 0, sinh u tan 0, tanh u = sin 0, tanh Ju = tan J^>, 

Now, if (p = amh i/ri, 

then cos ^ cosh i/ri = 1 , or cos0 cos *// = 1, 
a symmetrical relation, so that 

^ = amh 0/i ; 

and sin (p = tanh \fsi = i tan ^, 

cos (p = sech i//>i = sec \^, 
tan 0= sinh \fsi = i sin i/r, etc. 
Also d<p = i sech \lsid\fs = i sec 

A(0, ) = s/(! + ^ 2 tan 2 ^) = sec 
so that 

G.E.F. 



258 THE DOUBLE PERIODICITY 

If \/s = a.m(v, K ), 

then = am(w, /c); 

and sn(vi, K) = i *,., or isc(v, K), or HU(V,K); 



cn(vi, K) = TTXI or nc(v, * ); 
cu(v, K ) 

, . . . du(v, K) , , 

dn(m,*) = ^Yy or dc(v,*). 

connecting the elliptic functions of imaginary argument vi and 
modulus /c with the elliptic functions of real argument v and 
complementary modulus //. 

Putting v = K , we notice that sn K i, en K i, and dn TTi are 
infinite; and putting v = 2K , then 



also sn4 J ST / ^ = 0, en ^K i= 1, dn4^ i= 1. 

239. The Addition Theorems of 116 may now be written 
cn(u+ vi*) = (cuu cnv isn udnusn 
sn(u + vi) (snudiiv -\-icnudny. snv 
dn(u + vi) = (dn u en v dn -y ?/c 2 sn u en 16 sn v) -=- .D, 

jD = en 2 v + /c 2 sn 2 u sn 2 v ; 

remembering that the modulus of the elliptic functions of v 
is K, while that of the functions of u is K. 
Thus, putting v K t 

1 ,., .cnu 

= , dn(u+JTi)=-t --- : 

/csnu snu 



so that, putting u = K, 

w(K+ K i) = - IK IK, sn(K + K i) = I/K, dn(K + K i) = 0. 
Writing C, S, D for en 2u, sn 2u, dn 2%, then ( 123) 



Generally, when m and TI denote any integers, we find that 
cn(t6 + 2mK+ 2nK i) = ( - 1 ) m+n cn u, 
K i) = (-l) m sn u, 
2nK i) = (-l) n duu- 
so that 4fK and ZK i are the periods of sn u, 

2K and 4*K i are the periods of dn u ; 
the periods of cu u being 2(K+K i) and 2(KK i). 



OF THE ELLIPTIC FUNCTIONS. 259 

In 164, we may now write 

MI + ^2 + u s = 4wiK + 4<nK i ; 
or in the notation of the Theory of Numbers, 

^ + ^2 + ^3 = (mod. 4>K, K i\ 

240. A combination of the transformations of 29 and 238, 
to the reciprocal and to the complementary modulus, gives 

, . x 1 1 cnfc w, iic/ic) 

cn(m, K)= 7 - K= -3-7-7 v-/-/\= i / / , /( 

cn(v, K ) dn(r v, I/K ) dn(/c w, tc/* ) 

. ^_ 

^~ 



K dn( K vi, 



Thus cn^ w, i/c//cO = cd(w,, /c) = sn(E w,, AC), 

or am(/c u, //c/V) = i?r am(^T u, K) ; 

as is otherwise evident, when we notice that, if 

u =f (1 - * 2 cos 2 0)~*cty = ^/ (1 + ^ sinV) 



so that ^ = am(/c / u, i/c/V), 

then K-u=f(l- K -cos -\!s)~*d\!s= f (l-/c 2 sin 2 0)"^0, 

^ o 

or (j) = a,m(K U,K), 

provided i/r = JTT 0. 

241. As an application, take the values of v 1 and v 2 in 210 ; 

l + cos/3 cZ cos a cZ + 1 

2/ 2 



, v , 1 , _ 

1 1+cosa 1 + cosa 1 + cos a 

1 cos 8 o cZ cos a o cZ 1 . 

dnV 7 = 1 - --, sn 2 ^ 9 = ^ , cn-?; 9 = _- > 

1 cos a 1 cos a 1 cos a 

so that, with v l =pK i ) v. 2 = K+qK i, where p and q are real 

proper fractions ( 56), then 

1 cos a _ _ snH j _ sn~pl i dn z qK i 

1+cosa .sn 2 v cn 2 l i 



1 cos/3 KU Z V I dn 2/ y 2 _ K^stfpK i 

f+cos 8 == ~.sn 2 y dn = " 



d + 



200 THE DOUBLE PERIODICITY 

Thence, expressed in a real form, 
1-cosa = 
1 + cos a ~ 

or ( 135) tanJa- 
a= 

Also ( 29) 



so that = am{(2>+gX# , I//} -fam{(p -)/# , I//}. 

And 



or = 

In the Spherical Pendulum, O = ; and therefore ( 210) 
1 cos a 1 cos /3 d 1 _ -. t 
I+cosa l + cos/3 cZ + "l~ 

, c^ 1 

ancl 



/r/ -. z^/ 
en g/^ dn g^ 

or sn(p g)^T = sn pK cn qK du qK . 
Thence 

8n(g+y)JT cn(q+p)K 

" <)S/ ^- " ( " 



242. With Jacobi s notation of 189, the expression for i 
in 210 becomes 



-. 

- 

\ sn 



en 

-+ ZV,+ 

1 



snv 

and now, if we divide i/r into its secular and periodic part, 
i n the form \/r = ^fu/K + \/r , 

then ^ is called the apsidal angle, in the motion of the Top or 
of the Spherical Pendulum, as seen illustrated for instance in a 
Giant Stride and 



2 
which must now be expressed in a real form. 



OF THE ELLIPTIC FUNCTIONS. 261 

From 172, 

rvi 

iZ(vi, K ) = i/(dn 2 vi-E/K)dvi 



E fdn*(v, K) i 
= ^v/ - 9 ; , dv 
K J cv 2 (v, K ) 



E . t snvdnv , 

-K. en v 

sn v dn v 



7 ^ 
+ 



en v 

sn v dn i? 



by means of Legendre s relation of 171. 
Thus, with v l =pK i, 



Again, by (2)*, 186, since Z#=0, 

Z(^+ M.) = Zu - 
therefore, with v 2 = 



Also, if /> and ^ are proper fractions, the logarithmic term 
of i vanishes ( 264) ; so that, finally, 



In the Spherical Pendulum, 

n pK /sn pK = /c 2 sn pZ*sn qK si\(p q)K 



so that = 



With the Weierstrass notation, taking u in equation (8) 
of 208 between the limits o> 3 and ^ + 0)3, we find ( 278) 

i = (a 
where a = 

In small oscillations near the lowest position, p and K are 
very nearly unity, while q and K are small. 



262 THE DOUBLE PERIODICITY 

The Geometry of the Cartesian Oval. 

243. Denote the angles POO , POO, P0"0 in fig. 26 by 
0, 9 , 6" respectively ; then with as origin, 



_cu^w cn^w _ /A en n 1 cnw\ 
~~cn w+cn w ~ \ \l 



1 + cnw/ 
or, in a real form, with modulus K for the functions of v, 






... l/I 
~ \ \1 



cnu 1 cni>\ snjudnju 



en u en 



~ . n 

cos ^ = T , sin 9 = 



, . 

+ en u en v 1 + en u en v 

With /x as origin, 



and, similarly, 
^,_ 
~ 

_ //] 
~ \\l 



. ^,_ dn \w _ l/l duu 1 dn m\ 

~dnw + dnii/"V\l + dnu l+dnvi) 



sn 



M . n// 

cos = -T - , sin 0" = 



dn Ju 
/c 2 snusnv 



dni + dnucnt/ dn ^ + dn ucn v 

With as origin, and 

a; -j- yi = sn 2 Jty, 

,, i A/ 7 

then ^ tan W = 



sn 

To reduce this to a real form, similar to the above, we require 
two new formulas, not included in Jacobi s list ( 137), but easily 
derivable from it, namely, 



Now, with \w and \iv for u and v t and u and vi for 

and u v, 

dnu + cnu dnm 



// 
A/(j 

\\dnu-cnu dn 

,_ //dnu+cnu 1 dn v\ _ en |u dn \u sn |v en |t; 
~ V. Vdnu-cnu 1 + dn v/ " sn Jw dn Jv 

/c /2 sn u sn v 



cos 6> = 7- i sm (7 = , 

dnu cnudnv dnu cuuduv 



OF THE ELLIPTIC FUNCTIONS. 263 

244. Again, denoting the angles which P subtends at O O", 
0"0, 00 by 0, , <j>" respectively, so that 

= 7r -<9 -0", $ = 0-6", 0" = 7r-0-0 ; 
then we shall find 



tan i,_ sn i^ dn ^ cn i^ 


//I cnu l+cnv\ 


en tt sn Jv dn Jt 

. . /c sn Jit /c sn ^cn ^ 

for* i (4, 5 S ^ 


> Vu+cnit 1-cnv/ 
rV l/dnu cnu 1 dni>\ 


cn ju dn ^it dn %v 

. sn i& cn ^u cn ^v dn &v 


V \dnu + cnu 1+dnv/ 1 


dn Ju sn i ^ 


"A Vl + dnit dnv cn W 



cn u cn v . sn it sn v 

cos = ,- , sin d> = - , 

1 cnucnv 1 cnucnv 

, cn w + dn u dn v . , ic /2 sn u sn v 
cos0=-r ^ , sm0=- r - = . 

dn u + cn u an v dn it + cn u dn v 

cn v + dn u dn v /c 2 sn u sn v 

cos0 = j , sm0 =-= . 

dn t; dn u cn tr dn v dn % cn v 

Similarly, denoting by o>, a/, a/ the angles which the normal 
at P to the oval along which v is constant makes with PO, 
PO , PO", we shall find 

, sn u cn v sn it dn v . sn u 

tan o> = - , tan to = -5 , tan 



i . UCVXJL \JJ 

dn u sn v cn u sn v 

Drawing the three circles through O PO", 0"PO, OPO , and 
denoting the points in which the normal at P meets them 
again by Q, Q , Q", we shall obtain similar simple expressions 
for PQ, OQ, ... (Williamson, Liff. and Int. Calculus). 

245. The two ovals defined by v and IK v form a complete 
curve; and so also the ovals defined by u and 2KU. 

Denoting by P, P , Q, Q the four corresponding points 
defined by (u, v), (u, 2K -v), (2K-u, v\ (2K-u, ZK -v)\ 
and denoting by p, p, q, q their consecutive positions when 
u receives a small increment du, then 
Pp = tJJdu = K^/(rr r")du 

_ cn vi dn u + cn u dn vi //cn vi cn u\j 

dn vi + dn u \ Vcn m -f cn u) 
_dnu-fcnudnt> //I cn u cn v\ , 

dn ^ + dn it cn v \ \1 + cn u cn v/ 
and changing u into 2Ku, v into %K v, 

Q, f _ dn M- cn u dn w //I cn u cn t\ , 
dn v dn it cn t \ \1 + cn t/ cn v/ 



264 THE DOUBLE PERIODICITY 

TU 73^ i r\> > 2 dn^dnv //I cnucnt>\ , 

Inen rp + tyq =^ A/IT Jdu 

tc 1 + en w CD v y VI + en u en v/ 



-y - 2 ^(1 2 



so that the sum of the arcs described by P and Q is expressible 
as an elliptic arc. 



D r\> > /c /2 cn v , 

Again Pp Qq=^ -= ~\(T~, -cut, 

/c 2 1 4- en u en i> \ \l-f-cni6 en t> 

which is expressible in the form 
- 2 dn v cos 



2 
"*" 2. 2 >v/( dn2 ^ + 2 en V dn v cos 0" + cn 

so that the difference of the arcs described by P and Q is 
expressible by the sum of two elliptic arcs ; and thus the arc 
of the Cartesian Oval described by P is given by means of 
three elliptic arcs, which is Genocchi s Theorem (Annali di 
Matematica, VI., 1864 ; Mr. S. Roberts, Proc. L. M. S., III., V.). 

246. Let us examine the analytical properties and physical 
applications of the functions 

log en Jw, log sn ^w, log dn \w. 

Denoting log en \w by fa + ityv when resolved into its real 
and imaginary part, then 

= | log en %w en \w -f | log en J 



, en \w dn \w en ^w dn \w . _ , .en w r en 

= 9 lOff - ;; - - ^ - - - f - -- \~ 1 tan 1 



;; - - ^ - - - f - -- ~ - - 7 - j - 

dn Jto dn \w en \w + en %w 

en iv dnu + dnmcnu ., ,. Il cnu 1 cnm 



as in 236, by means of formulas (3), (20), (28) of 137 ; and 
now expressing the elliptic functions of vi t to modulus /c, in 
terms of functions of v, to modulus /c understood ; then 

_il ^dnu + cnudnt; _ 1 l/lcnu 1 en tA 

^ l ~ to dn v + dnucn f ^~ \\l + cni6 1 

Denoting logsn \w by (f> 2 + i\ls 2 , then 
= J log sn Jwsn Jty + J logsn J 

, sn Aw dn iit; sn A^ dn itt; . . , .sn \w sn 
log ^ -4-^ - hitan-H f y 

dn %W dn J^ sn \w + sn 

cnm en u , .. ,. //dnu-fcntt dnvi cnvi 
~ 



,. //dnu-fcntt dnvi cnvi\ 
~H A /(-i --- -r ^ :) 
\\dnu en u dnvi-fcntrt-/ 



_j, 
~ 



OF THE ELLIPTIC FUNCTIONS. 265 

1 cnucn-y 



2 cn 



. _ 1 //dnu + cnu 1 dnv\ 
v \\dnu-cnu 1-fdn-y/ 



Similar!} 7 , denoting logdn^iv by fa + i fiy it 

cuviduu + cuuduvi . .. //I -dim l- 

~ 



: A . 

en w + en u \\l + dnu 1-fdnw/ 

idnu dnv c 



l/ 
1 */! 

\ \ 



1+cnucnt; \\l-r-dni6 

By (20), (21), (22), (23) of 137, we prove, in a similar 
manner, 

l + cn-it . cuvi-\-cuu , , A _, isuvidiiu 
1 emu 2 cnvi en 
= tanh -1 (cn u en 

= tanh ~ J (dn u en v/dn v} i tan ~ 1 (cn u sn v/sn u), 

y ) = etc. 



i w en w 

247. These conjugate functions ^ and -^ of the complex 
u-\-vi are capable of representing the solution of various physi 
cal problems concerning a plane in which u and v are taken as 
rectangular co-ordinates, since they satisfy the conditions 

3u ~dv *dv 3i6 



Here tt and v are not restricted to be rectangular co-ordinates, 
but they may represent the conjugate functions of confocal 
conies or Cartesian Ovals, as in 179, 236, or of any orthogonal 
system, which divides up a plane into elementary squares or 
rectangles, as on a map or chart. 

As in 54?, we take a period rectangle OABC, bounded by 
u = 0, u=2K, v = y v = 27i ; and now, as the end of the vector 
w or u + vi, drawn from 0, travels round the boundary OABC 
of this period rectangle, the vector w assumes the values 
ZtK(Q <*<!); 2#+ 2* / JGT i(0 <$ <!); 

2tK+2K i(I > t > 0) ; 2 JTi(l >t > 0). 

When the sides of the period rectangle are a and 6, we 
replace u and v by 2Kx/a and 2K y/b, where K /K=b/a. 



266 THE DOUBLE PERIODICITY 

Taking the function log en \w or ^-}-i^ v then from to A, 
V^i = ; from A to B, ^ = ITT ; from to (7, ^1=2^; an d fr m 
G to 0, ^ = 0. 

At A, where K = 2^T, -y = 0, then 1= = oo ; and at (7, where 
u = 0, -y = 2K, ^ = oo . 

The functions ^ and i/r lf therefore satisfy the conditions 
required of the potential and stream function, due to electrodes 
at A and C, of the plane motion of electricity or fluid, when 
bounded by the rectangle OABC. 

The function ^ will also represent the stationary tempera 
ture at any point of the rectangle, when the sides OA, OC are 
maintained at temperature zero, and the sides AB, BG at 
temperature JTT. 

When the period rectangle is a square, or KK\ then 
\^i- i 71 " when u+v = 2K, or along the diagonal AC; we thus 
obtain the permanent temperature inside an isosceles rect 
angular prism, when the base is maintained at one constant 
temperature, and the sides at another. 

Similar considerations will show that the function logsnjw 
or 02 + ^2 w iH &i ye ^ e streaming motion in the same period 
rectangle, due to a source at 0, and an equal sink at C. 

The function \js z is now zero along OA, AB, EG, and JTT along 
0(7; and ^ 2 will therefore represent the stationary temperature 
when OC is maintained at temperature JTT, while the other 
sides are maintained at zero temperature. 

A superposition of four such cases will give the permanent 
temperature when the sides of the period rectangle are main 
tained at any four arbitrary constant temperatures. (F. Purser, 
Messenger of Mathematics, VI., p. 137.) 

EXAMPLES. 
1. Solve the equation 



2. Investigate the curves given by 

dz/dw = (I-z*)$. 

3. Prove that the system of orthogonal curves given by 



are the stereographic projections of a system of confocal sphero- 
conics ( W. Burnside, Messenger of Mathematics, XX.). 



OF THE ELLIPTIC FUNCTIONS. 267 

Prove that the stereographic projection of the points 



on the sphere 

whose latitude and longitude are 6, <p, are given by 



Prove also that 

^ 



toy f%y /as y _ /^y (*y\* n* 

va) + v^J ^ \d~J ~\^v) + w + W 



4. Discuss the physical interpretation of 

-ic/c sn u sn v . .ic cnv . 



dn u dn -y /c en u 

and determine the single function from which it is derived ; 

K sn u sn v 



f , . . . , 

also of 0-h^0 = tanh~ 1 J +^tan 



J 

dn n dn v en v 

Interpret these expressions when 



5. Prove that, if x + yi = sn w, 
then 



gives the plane motion of liquid streaming past two obstacles 
given by x = l and l//c, x= 1 and l/ K (W. Burnside, 

Messenger, XX.). 

27ie Double Periodicity of Weierstrass s Functions. 

248. A procedure similar to that of 236 will show that the 
Cartesian Ovals of fig. 26 are also the representation of the 
conjugate functions of the system z ^w, obtained from the 
definition of 50, 

r dz 



or dz/dw = p w=- 

where 4s 3 -g^-g 3 = 4(2 - e^(z - e. 2 }(z - e a ) ; 

and z = e lt e 2 , e B define the three foci. 
According to 51, 



&iv -e z = (e l - 6 3 )ds V(^i ~ ^)w = (e 2 - e 8 ) cn 
pw - 6l = (e x - e 3 ) cs V^ - e^tv =-(e l - ^ 3 )dn 
by 239 ; thus identifying these results with those of 236. 



268 THE DOUBLE PERIODICITY 

With the notation of 202, 



and denoting the focal distances by r lt r 2 , r 3 , and u vi by w , 



249. To express these focal distances in a real form, as in 236, 
we employ the Addition Theorem (K) of 200, written 
a-(u + v)v(u v) = o-% cr 2 f { (pv e a ) (<pu e a ) } 

= (r 2 uo- a -v-cr a 2 u ( r 2 v .................... (M) 

Again, from 154, $>(u + v) e a is a perfect square; and we 
may write x = pu t y = $>v, s = p(u+v) > 



_ -eg.pv-ep.pv- e y ) - */(pu-ep . pu-e y . $>v-e a ) N 

$>v $>u 
and now 

<r a (w + v)a-(u v) = +J { p(u + v) Co] cru ar 2 v (<pv <pu) 

= VU <T a U (TpV (T y V <TpU <T y U (T^V (TV,... (0) 

and changing the sign of v, 

ar(u + v)a- a (u v) = o-U (r a u oyy o- y v + cr^U o- y u a- a v crv. . . .(P) 
Again, by multiplication with (N) and reduction, 
v) 



or 

<r a ( u + v )<rp( u ~ v) = ar a u a-pU a- a v a^v - (e a - e^a-u a- y u <rv <r y v, (Q) 
cr a (u v)arp(u + v) = o- a u a-pU a- a v a-pV -f (e a -e^)a-u a y u crv cr y v. (R) 
Similarly, 

(u -v) = (pu- e a )(^v - e a ) - (e a - e ft )(e a - e y ) 



cr(u--v) <pv <pu 

or 
<7 a (u + v)o- a (u - v) = (r a ~u cr a 2 v - (e a - 6p)(e a - e y )ar 2 u <r*v ....... (8) 

(Schwarz, Elliptische Functionen, p. 51.) 



OF THE ELLIPTIC FUNCTIONS. 269 

Now, from these equations (0), (P), (Q), (R), with 

w or %(u+vi) for u, and w or ^(u vi) for v, 
_ a-w a-iv <r<tf(r.v , vii (rvi 



or r = 



with similar equations for r 2 and r 3 ; and thence the vectorial 
equations of the Cartesian Ovals analogous to those of 236 

r z <r s u - r B a- 2 u = (e. 2 - e^u \ etc ^ 

r 2 o- 3 t i r 3 <r. 2 v i = (e 2 ejvjvij 

These vectorial equations again are the geometrical inter 
pretation of the formula, immediately deducible from (N), 



(T) 

Making m 2 = 1 in the homogeneity equations of 196, gives 

V( 5 92 &) = ~ V( v 5 02> -0s) 
the equivalent of the equations of 238, by which a change is 

made to a real argument and complementary modulus ; while 
(w; 02> 3 )= - 4(^5 02 -9s) 

-(vi , 2 . 3 )= ^>; 2 . -0 3 ) 

o- a (^; 2 3 )= <r a ( v > 02 -0s)- 

250. When a point has made a complete circuit of one of the 
ovals, enclosing a pair of foci, defined by e. 2 and e 3 , or e l and e 2 , 
z will have regained its original value, but w will have increased 
or diminished by 2^ or 2o> 3 , defined as in 51, 52 by the 
rectilinear integrals 



so that 2^!, 2o) 3 are the periods of the function pw, and 



To fix the ideas we have supposed the circuit of two poles 
of the integral made on the enclosing branch of a Cartesian 
Oval, but the result will be the same whatever be the curve, 
provided it makes the same number and nature of circuits. 

Now, in 165, we can have 

EE() (mod.20)!, 2o> 3 ). 



270 THE DOUBLE PERIODICITY 

251. In 54 it has been shown how, as the vector of the 
argument w traces out the contour of the period rectangle, <pw 
assumes all real values : and $w may be made to assume any 
arbitrary complex value at a point in the interior of the 
rectangle, given by a determinate vector t^ + t w B . 

It is convenient to put o) 1 + a) 3 = o> 2 , so that 

ft>! + w 2 + w 3 = 0, with gj_ + 6 2 + e s ; 
and now ^o^ = e v $a) 2 = e 2 , #>o> 3 = e 3 ; 

while %> u>i ~ fi coz = $> to 3 = 0. 

The equations of 54 show that 

e-. e 9 . 6, Co 



A 

equations analogous to those of 57, in Jacobi s notation. 
Thus, from ex. 9, p. 174, 



With negative discriminant, as in 62, we take e 2 as real, 
and e v e 2 imaginary; also o^ = J(w 2 + a/ 2 ), o> 3 = J( W 2 ~~ w/ 2) > 



252. A great advantage of the Weierstrassian notation (at 
first rather baffling to one accustomed to the methods of 
Legendre and Jacob!) is that the dimensions of the elliptic 
integral are left arbitrary, and can be changed by an applica 
tion of the Principle, of Homogeneity of 196. 

When the canonical elliptic integral of 50 is normalized 
in Klein s manner ( 196) by multiplying by A T \ then 
A^eZs r da- 



where s = AV, (/ 2 = A^y 2 , g 3 = 

and now y> 3 27y 3 2 = 1, 

so that the new discriminant is unity, and 

T O T" 1 \*w O 



If GT P trr 3 denote the real and imaginary half periods of the 
normalized integral, then 



= ft) 3 



OF THE ELLIPTIC FUNCTIONS. 971 

The general elliptic integral, written with homogeneous 
variables as in 155, is also normalized by Klein by multiply 
ing by the twelfth root of the discriminant of the corresponding 
quartic, and its half periods are now c^ and C7 3 . 

If we normalize, for instance, the canonical integral (11) of 
38, written with homogeneous variables x v X 2 , in the form 

f(x^x. 2 .x 2 x l .x. 2 kxjrtyfyfa^ x l dx. 2 ) > 
then the invariants g z , g 3 , and the discriminant A of the quartic 

12 * 2 ^~ 1 " *^"> ~ A/tX/-fl y 

being the expressions given in 68, therefore 



Now the half periods of integral (11), 38, being 2K, 2K i, 



We are thereby enabled to change from Weierstrass s (a l and 
o> 3 to Jacobi s K and K, and to utilize the numerical results of 
Legendre s Tables. (Klein, Math. Ann., XIV., p. 118.) 

When the discriminant A is negative, we normalize by 
multiplying by ( A) 1 ^, and replace o^ and o) 3 by u> 2 and o>./ 
( 62); but now the new discriminant y 9 3 27y 3 2 = 1, and 
o>. 2 ( - A)" = SA^/Q/o/), o/,( - A^) = 2K i 4/(^ ) ( 47, 58). 

For instance, if g 2 = in 50, (-A)^=^/3^/(/ 8 ; and in 58, 
,7=0, or 2/C/-J, 24/(J^ ) = 4/2; and now 



while ( 47) wjVwj = K ilK= i 

Confocal Quadric Surfaces. 

253. The symmetry and elegance of the Weierstrass notation 
is well exhibited in the physical applications relating to con- 
focal surfaces of the second degree. 

The equation of any one of a system of confocal quadrics 



we put 

a 2 + \= 
and now the interal 



d\ 



X 

With e l >e. 2 > e 3 , we must take a- <b 2 < c 2 . 



272 THE DOUBLE PERIODICITY 

Three confocals can be drawn through any point x, y, z, 
an ellipsoid, a hyperboloid of one sheet, and a hyperboloid of two 
sheets. 

Supposing the ellipsoid to be defined by X or u, and the 
hyperboloid of one sheet in a similar manner by fj. or v, and 
the hyperboloid of two sheets by v or w ; then in going round 
the period rectangle of 54, 

(i.) u =pco, oo > pu > e v for the ellipsoids ; starting with p = 
for the infinite sphere, and ending with p = \ for the inside 
of focal ellipse; 

(ii.) v = coj + <?ft>3, e l >pv>e 2 , for the hyperboloids of one sheet; 
starting with q = from the focal ellipse, and ending with 
(7 = 1 for the focal hyperbola; 

(iii.) W = ?*! + o> 3 , e 2 >$w>e s , for the hyperboloids of two 
sheets ; starting with q = l. from the focal hyperbola, and ending 
with q = for the outside of the focal ellipse ; 

(iv.) the fourth side of the period rectangle gives imaginary 
surfaces. 

254. Replacing 6 2 -a 2 and c 2 -a 2 by /3 2 and y 2 , so that 



are the equations of the focal ellipse of the confocal system, we 
should have to put, with Jacobi s notation, 
= 7 2 cs 2 (u,,c), tf + \= 



, 
where 



and now u, v, w will be Lame"s parameters, as given in Max 
well s Electricity and Magnetism, I., chap. X. 

By solution of the three equations of the confocal quadrics, 



_ 



c 2 ct 2 . c 2 6 2 

and thus x, y, z can be expressed as functions of u, v, w. 
Employing the function s a of 203, 



ms 



22 



, y 2 = 



OF THE ELLIPTIC FUNCTIONS. 273 

When 6 2 = c 2 , the ellipsoids are oblate spheroids, and the 
hyperboloids of two sheets degenerate into planes through Ox ; 
and now the orthogonal system is given by 



1 V 2 


(i } 


cot% cec 2 u / 


w 

^ii ^ 


y 2 z 2 
y n - 


\ u -) 

...(ui.) 



intersecting in the point 

x = y cot u tanh v, 

y = y cec u sech v cos w, 

z = y cec u sech v sin w. 

When b 2 = a 2 , the ellipsoids are prolate spheroids, and the 
hyperboloids of one sheet are planes through Oz\ now the 
orthogonal system is given by 



r~ + 



cechhi coth% 



+ 



sechhv tanh% ~ ^ 
intersecting in the point 

x = y cech u sin v sech w, 
y = y cech u cos v sech w, 
z = y coth u tanh m 

The degenerate case of confocal paraboloids, where the centre 
is at an infinite distance, may be written 
2/ 2 2 



y 2 



(viii.) 



intersecting in the point 

x = a(cosh u + cos v cosh w\ 
y = 4ta cosh J^ cos ^v sinh Jw, 
= 4a sinh ^u sin ^v cosh ^w. 

(Proc. Lond. Math. Society, XIX.) 



Q.E.F. 



274 THE DOUBLE PERIODICITY 

255. We may take u, v, w as Lame s thermometric para 
meters, and now Laplace s equation becomes (Maxwell, Elec 
tricity, I., chap. X.) 



Thus <j) = 

+ 2Evw + 2 Fwu + 2 Guv + Huvw 

is a particular solution of this equation; for instance, the 
electric potential between two confocal ellipsoids, defined by 
U-L and it 2 , maintained at potentials U-^ and U% is given by 



When the solution <^> is equal to UVW, the product of three 
functions, U a function of u only, F of v, and W of w only, 
then Laplace s equation becomes 



ft 

= 

so that we may put 



three equations of Lame s form ( 204), when g = 

256. The complete solution of Lame s equation was first 
obtained by Hermite, in the form 



Denoting by Fthe product U^U 2 of U^ and U 2 , or F(u) and 
F( u), two particular solutions of the general linear differential 
equation of the second order, in its canonical form 



where 1 is some function of u, and denoting differentiation 
with respect to u by accents, then 



or F / -2/F=2C7 1 / C7 2 ; 

and F //x - 2/F - 2/ r F- 2 [7/ CT 2 + 2 Z7/ f// 



or F // -4/F / -2/ / F=0, 

the general solution of which linear differential equation is 



OF THE ELLIPTIC FUNCTIONS. 275 

A first integral of this differential equation is 

2 YY" - F 2 - 4/F 2 + C 1 = 0, 
where C is a constant, given by 

U& -UJU^C, 

the integral of ^ J7 2 " - Uf U. 2 = 0. 

In Lame s differential equation 



and now, changing to x = pu as independent variable, 



and this equation for Fhas, as a particular solution, a rational 
integral function of x or jm, of the -nth order, which we may 
write F= 

and ^ 



Now, by logarithmic differentiation, 



?7 2 ^ F 

Brioschi shows (Comptes Rendus, XCIL) that, when resolved 
into partial fractions, we may put 



pa) 
provided that 



and 
Then 



and, integrating, 

Fu, or [T, = n e X p( - fa) = II0(, a) ; 

while Z7 2 or J^ u) is obtained by changing the sign of or a, 



276 DOUBLE PERIODICITY OF THE ELLIPTIC FUNCTIONS. 

257. Hermite shows (Comptes Rendus, 1877) that the func 
tion F(u) may be otherwise expressed by 

/ /I \ n-I 

*()=( 



and <f>u, called the simple element, is of the form e Xu cj)(u, w), 
</>(u, co) being a solution for n = 1 and h = poo ( 204). 

To obtain the coefficients A v A 2 , ... in F(u), we suppose 
<u or e Xu (p(u, (o), JF u, ^?u expanded in the neighbourhood of 
u = Q ( 195), in the form (Halphen, F. E. /., chap. VII.) 



Substituting in Lame s differential equation 

F"u= {n(n + I)pu + h}Fu, 
we obtain, by equating coefficients, 



_---- 

- 10 



On comparing the two forms of the solution Fu, we find that 

w = 2a, and X = fo> 2fa. 
Thus, for instance, when TI = 2, we find, as in 209, 



d j( _ f(( _ 



When 7i = 3, 



w h e re ctj + a 2 + a 3 = a>, 



r l liis fails when g 2 = 0, and a 1 = v, a 2 = wf, a 3 = aj 2 ^ ; but now 
(229) J^ = i(^ 



CHAPTER IX. 

THE RESOLUTION OF THE ELLIPTIC FUNCTIONS 
INTO FACTORS AND SERIES. 

258. The well-known expressions for the circular and hyper 
bolic functions in the form of finite and infinite products 
(Chrystal, Algebra, II., p. 322; Hobson, Trigonometry, chap. 
XVII.) have their analogues for the Elliptic Functions, as laid 
down by Abel in Crelle, 2 and 3. 

Granting the possibility of the resolution into linear factors, 
the individual factors are readily inferred from a consideration 
of the zeroes and infinities of the function. 

Denote 2mK+2nK i by ft, 

where m and n denote any integers, positive or negative, 
denote also Q + K or (2m + I)K+ ZnK i by Q v 
tt + K+K i or (2m + l)K+(2n+l)K i by Q 2 , 
and Q + K i or 2771^+ (2n + l)K i by Q 8 . 

Then considering the function 

sn u, 

the zeroes are given by u = fi, and the infinities by u = Q 3 
( 239) ; and thus we infer that, if sn 11 can be resolved into 
a convergent product of an infinite number of linear factors, 
the form is 

m = oo n = oo / n, \ 

u IT IT (l-j) 

" =-""=-< X (1) 



the accents in the numerator denoting that the simultaneous 

zero values of m and n are excluded. 

277 



278 THE RESOLUTION OF THE ELLIPTIC FUNCTIONS 

Similarly, cnu = BTLn(l-) /D, .............. . ...... (2) 

...................... (3) 



the zeroes of cnu being given by u = l v and the zeroes of 
dnu by u = } 2 , while the infinities are given as before by 
u = Q 3 ; D denoting the denominator in (1). 

259. But now, in demonstrating the analytical equivalence 
of the expressions on the two sides of equations (1), (2), (3), it 
will fix the ideas if we employ a physical interpretation, such 
as that given in 247. 

It was shown there that the real and imaginary part (norm 
and amplitude) of 

log sn w, 

where w = u + vi, will represent in the rectangle OABG the 
potential and current function of the flow of electricity (or of 
liquid, following the laws of electrical flow) from a positive 
electrode at to a negative electrode at C, JTT amperes being 
the strength of the current ; but here we take OA=K, OG=K f ; 
and u, v are the coordinates of any point in the rectangle. 

The infinite series of electrodes, which are the optical images 
by reflexion of these two electrodes at and (7, will form a 
system on an infinite conducting plane, such that, if the 
strength of the current at each electrode is 2?r amperes, the 
resultant effect in the rectangle OABG will be the same as 

before. 

(Jochmann, Zeitschrift fur Mathematik, 18G5; 

0. J. Lodge, Phil. Mag. 1876 ; Q. J. M. y XVII.) 

Starting with a single electrode at 0, of current 2?r amperes, 
the potential and current function at any point whose vector 
is w or u + vi are the norm and amplitude of logw; and logiv 
may be called the vector function of the electrode at 0. 

For an electrode at a point whose vector is c a-\-bi, the 
vector function 8&z=X+yi is log(0 c), 
which may be written 



disregarding the complex constant log(-c). 



INTO FACTORS AND SERIES. 279 

The vector of any optical image of in the sides of the 
rectangle OABC being given by Q, the vector potential of the 
corresponding electrode is log(l w/Q); and the vector function 
of the system of images of the positive electrode at will be 



Similarly the vector function of the system of images of the 
negative electrode at C will be 



But these functions, considered separately, represent a 
physical impossibility, and are analytically meaningless ; their 
difference, however, 



will represent the vector function of the whole system of posi 
tive and negative electrodes ; and since this function satisfies 
the requisite conditions inside the rectangle OABC as the 
function logsnw, we are led to infer equation (1), with suitable 
restrictions explained hereafter. 

For log en w, the positive electrode is placed at A, the 
negative electrode being still at (7; the vectors of the positive 
electrode images are given by Q : ; and now equation (2) is 
inferred ; while for log dn w, the positive electrode is placed 
at B, and the vectors of its images are given by Q 2 , the 
negative electrode being at (7; and we infer equation (3). 

When in the rectangle OABC we have OA = a, OC=b, 
we take K lK^bja, and write K(x/a) + K i(y/b) for u + vi* 
x, y now denoting the coordinates of a point. 

260. We now proceed to express these doubly infinite pro 
ducts of factors, corresponding to the different integral values 
of m and n, by means of singly infinite factors for different 
values of n ; that is, we combine all the factors for one value 
of n and the infinite series of values of m into a single ex 
pression; and here we employ the formulas for the trigono 
metrical functions expressed as infinite products. 

Interpreted physically, we determine the vector function of 
an infinite series of electrodes, equispaced on a straight line 
parallel to OA. 



280 THE RESOLUTION OF THE ELLIPTIC FUNCTIONS 

Denoting the vectors of such a series of positive electrodes 
by 2ma+nbi, the vector function is 

m= / _ _ 

log n (z-2ma-nbi), or iog(-n6i)IT(l- 



and provided that (z nbi)/2ma is ultimately zero when m is 

infinite, or that z/ma and n/m tend to the limit zero, we can 

write this vector function (Cayley, Elliptic Functions, p. 300) 

log sin \-w(z nbi)/a, ....................... (4) 

Resolved into its norm and amplitude, this vector function is 
i log J[cosh{7r(2/ nb)/a} cos TTX/O] 

+ itan~ 1 [tanh{j7r(2/ ti&)/a}cot(j7ra;/a)]. ...(5) 

The amplitude or current function is therefore constant when 
a3=(2m-hl)a; and there is no How across these lines, provided 
however, as is physically evident, we do not recede to such a 
large distance from the origin that we are not justified in 
taking It z/2ma as zero. 

261. We suppose that Oy passes through the centre of this 
infinite series of electrodes, or that m reaches to equal infinite 
positive and negative values; but now, at a very large dis 
tance from 0, the electrodes on one side of a line, given by 
o; = (2m-|-l)a, where m is a large number, will preponderate 
over the electrodes on the other side, and the resultant effect 
will be a uniform normal flow a across this line, to counteract 
which a term of the form az or loge~ az must be added to the 
vector function. 

The analytical equivalent of this physical effect is illustrated 
by the theorem proved in Hobson s Trigonometry, p. 328, that, 
when the integers p and q are made infinite in any given 
ratio, then 00, the limit of the product 



Ji) ... (1+ 

w V 



2aA a/ \ a/\ 2a/ \ qa 



The infinite product TL(l+c n x) is convergent for all finite 
values of x, if the series 2c n is convergent ; as is evident on 
expanding the logarithm of the product. 



INTO FACTORS AND SERIES. 281 

But Weierstrass shows (Berlin Sitz., 1876) that the divergent 
product 

can be made convergent if the exponential factor e z l ma is 
attached to the linear factor Iz/ma; or, interpreted electri 
cally, if to the motion due to the electrode at ma, whose 
vector function is log(l z/ma), we add a uniform streaming 
motion parallel to the vector ma, given by log e z l ma or z/ma. 
Now, denoting the harmonic series 



since the limit of s p log_p or s q logq is Eulers constant. 

262. In a similar manner it is inferred that the vector 
function of an infinite series of positive electrodes, whose 
vectors are (2m + l)a-f-7i6i, 

m reaching to equal positive and negative infinite values, is 
log cos %Tr(z-ribi)/a = JlogJ[cosh{7r(y-w&)/a} + cos(7r#/a)] 

+ i tan- 1 [tanh{ %Tr(y-nb)/a}t&n(^7rxla)], (7) 
having lines of equal amplitude given by x = 2ma. 

Therefore the vector function of a pair of lines of electrodes, 
whose vectors are 2manbi, is 

log sin{ \TT(Z nbi)/a }sin{ \ir(z -f nbi)/a} 

= log J{cosh(7i7r&/a) cos(7r^/a)} ; 

or, corrected by the addition of a constant, which makes the 
function vanish when z = Q, the vector function is 
, cosh(ti7r&/a) cos(7T0/a) , 1 2o w 





where q = e- 7rb/a . 

For a pair of lines of electrodes whose vectors are 
(2m + l)a?i6i, the vector function is 



which may be replaced by 

, cosh (ri7r6/a) + cos (irz/a) , 1 + 2q n cos(7rz/a) + (? 2n /0 ^ 
cosh(7i 7 r6/a)-|-l W2 



For the line of electrodes along OA, whose vectors are 2ma 
or (2m + l)a, the vector function will be 

log sin(j7r/a) or log cos(|7T0/a) ................ (10) 



282 THE RESOLUTION OF THE ELLIPTIC FUNCTIONS 

263. Under Cayley s restrictions, that m reaches to equal 
positive and negative infinite values, and n also ; but that the 
infinite values of n are infinitely small compared with the 
infinite values of m (equivalent to taking the infinite array of 
the images of the electrodes as contained in an infinite rect 
angle, of which the length in the direction OA is infinitely 
greater than the breadth in the direction OB), we can now 
replace the doubly infinite products in (1), (2), (3) by singly 
infinite products, in the form 



, 

sn u = A sin(j7ru/A) II - r- v +D, (11) 

71-1 I 1 "*? ) 



en u = B cos( JirM/JSr) II __L S -_ 4-D, (12) 

dnu= Oil 

where 

D= n 



By putting u = 0, the values of A, B, G are seen to be 

, 1, 1 ; while g = exp( - ^K \K}. 
The common denominator D of the three elliptic functions, 
which represents physically a function whose logarithm is the 
vector function of the negative electrodes at points whose 
vectors are of the form Q 3 , is the equivalent of Jacobi s Theta 
Function of 187; and we write 
1 



-00 nl 4 

Li + 



The numerator of sn u will now be the equivalent of the 
Eta Function, defined in 192 ; and thus 
Hu = K sn u Qu 



. ...(16) 



The numerator of en u is represented by the Eta Function 
of u + K, and the numerator of dn u by the Theta Function of 
and the factors are so chosen that 



INTO FACTORS AND SERIES. 283 



Equation (6) of 188 may now be written 

while, by means of (7), 137, 

H(+t;)H(u-t;)8*0=Hue 8 -eVBP (19) 

264. It is convenient to replace \iru\K by a single letter x ; 
and we shall now find that the constant factors are so adjusted 

as to give the expansions in a Fourier series in the form 

o 9 a i /o/"^\ 

cin R/y {91\ 

sinox t^i; 

It is easily shown algebraically that 

71=00 

n=l 

by changing z into <fz and multiplying by qz, when the pro 
duct on the left hand side merely changes sign ; whence equa 
tion (20) is inferred from (15) by putting z = e 2xim } and equation 
(21) is obtained from (20)* by writing qz for z, and multi 
plying by q^z*. 

Written in the exponential form, 



>* (22) 

or with g = e~ a , a = 7rK IK, arid b = xi, 



n -^ b - ...... (24) 

and e(u+2K)= Ou, 

H(u + 2A r )=-Hu, .................... (25) 

Changing u into u + K i, or a; into x + ^i\ogq, we find 



iq-le-^eu, .............. (26) 

agreeing in giving K snusn(u + K i) = l, .................... .(27) 

and leading by differentiation to the formula 

Z(u + K i) = Zu + (cnuduu/suu)-(i7rilK), ......... (28) 

which, with ( 176), 

Z(u+K) = Zu-(K 2 snucuu/dnu), ..................... (29) 

leads to 

nu/cnu)-(%Tri/K) .......... (30) 



284 THE RESOLUTION OF THE ELLIPTIC FUNCTIONS 



265. Jacob! writes (Werke, I., p. 499) x for ^iru/K, and 
Ox for 9tt, OjX for Hit, ^ for H.(u + K), and # 3 ce for Q(u+K) , 
and now 



......... (31) 

m ox ......... (32) 

m*if*-1W*~V* 

= 2q$cos x + 2^ cos %x -f 2^cos ox + ......... (33) 

= 2^V na:i 

......... (34) 



or, with q = e~ a , b = xi, 
Ox = Ei 2n exp( - 
O^x = 2 exp ( 



(35) 

Conversely, starting with these functions as defined by 
these exponential series, it is possible to rewrite the whole 
theory of Elliptic Functions ab initio in the reverse order, and 
to deduce all the preceding results. 

(Jacobi, Werke, I., p. 499 ; Clifford, Math. Papers, p. 443.) 
For instance, we find that 

6(x + JTT) - flgOJ, 0(x + \i log q)=- iq^e^x, 
O^x + JTT) - 2 x, O^x + \i log q) = - iq-$e xi 6x, 
2 (x + ITT) = - 0^, 6 z (x + Ji log ? ) = g-V flgB, 
3 (x + JTT) = 0c, 3 (o; + Ji log q) = q-iePOjc ...... (36) 

The quotient of two functions is thus a doubly periodic 
function, of reaZ period 2?r or TT, and imaginary period ilogq. 
The form of the and function series shows that they 
satisfy partial differential equations of the form 



_ 

dx z d log q 

and the functions are therefore suitable for the solution of 
problems in the Conduction of Heat. 

Thus, if 6(x cos a + y sin a, q) represents at any instant, = 0, 
the temperature at the point (x, y) of an infinite plane, of 



INTO FACTORS AND SERIES. 285 

which y denotes the ihei^mometric conductivity, then at any 
subsequent time t, the temperature will be given by 

0(o; cos a + # sin a, qe -W) .................... (38) 

266. Similar considerations to those of 258 enable us to 
resolve other expressions into factors ; for instance, 

., . , dnu-f/ccnu 
, or its reciprocal 



so that 



K K 

dn u KCIIU K /dn u K en u 



Vdnu K 
dn u -f- K 



_ A/ , 

K dn u + K en u \ dn u + KCHU 

Now dc u, or sii(Ku) = l//c, when 



or cos \TTUJ K= cosh(2r& - 1) 

while dc u = I/K, 

when cos %TTU/K= cosh(2w V)^ 

and therefore we may put 



dn u K en u _ _.cosh(2?i V)\TrK ]R cos \irU\K 



, 

11 n - -" - 1J 



where the letter C is used to denote some constant factor. 
Now, writing x for ^TrujK, and supposing x and it real, 
log(l - 2c cos x + c 2 ) = log(l - ce xi ) + log(l - ce-**) 

= 2(c cos x + Jc 2 cos 2a; + Jc 3 cos 3x +...), 
log(l + 2c cos cc + c 2 ) = 2(c cos x Jc 2 cos 2ic + Jc 3 cos 3a: ...), 



log = - -5 = 4(c cos a; + ic 3 cos 3x + |c 5 cos bx +...). 

1 + LC cos x -f- c - 

Therefore, expanding the logarithm of (39), 
log- l ^- 
= logC- 

= \ogC- 



,l-q~ 3 1-53- 5 !_ 5 5 

yv AV 1 cos(2??i l)^7ru!K 
= \ogC 22 r- -. , ^ , 

and, differentiating, 



snu=S , ,..(41) 

JL sinh2m 



the expression of sn u in a Fourier Series. 



286 TH E RESOLUTION OF THE ELLIPTIC FUNCTIONS 

267. By forming the similar factorial expressions for 

Ksnu+iduu and suu + icnu, 
and taking logarithms, we shall find 

log(/c sn u + i dn u) 

.^, 1 sin(2m 
= constant SiS 



2m -1 eosh(2m 

, , , . N .^1 sin rri 

log( sn u+^ en u)= constant ^E r t, , r ~ ...... (43) 

m cosh m-TrK IK 

and, differentiating, 

?r^ cos(2m 



TT 



and therefore, integrating, 



We have now found that, in 78, 

1 



n cosh 
268. From 263, we find, in a similar manner, that 






7T 2 



Now, referring back to 78, we can put 

^ 7T 1 7T 2( 



TTX _, sn - 

am u = -x-^4- 2 r -- WTTT ................ (46) 

m cosh m-jrK IK 



v l c 
= constant 2, -- ^TT -- 7^771^; .................. (47) 

m Binh(mxJi 7 JT) 

and, differentiating, 

7r s ^iKm7rV^_ 
-^ J " 



(49) 



K Kl-q^ 
Putting u = in (49) or (50) gives what is called "a q series," 



m toiuf* _K(K-E) 

//77-\ ^ i 



INTO FACTORS AND SERIES. 287 

As an exercise, the student may form the similar factorial 
expressions for 

1 cnu 1 snu 1 dnu duu cnu 
etc. 

sn u en u- K sn u yc sn u 

and their reciprocals 

1-fcnu 1 + snu 1 + dnu dnu + snu , 

sn u CD u K sn u jc sn u 

and thence determine, by logarithmic differentiation, the Fourier 
Series for ns u, cs u, ds u, etc. (Glaisher, Q. J. M., XVII.). 

The applications of these expansions will be found in papers 
in the Q. J. M., XVIIL, XIX., XX. 

269. As an application of these q series, consider the problem 
of the electrification of two insulated spheres, in presence of 
each other, of radii a and b, and at a distance c from centre 
to centre, when maintained at potentials V a and V b , with 
charges of E a and E b (Maxwell, Electricity and Magnetism, 
I, chap. XL). 

Then E a = q aa V a + q ab V b , E b = q ab V a + q bb V b , (52) 

where q^, q bb are called the coefficients of capacity, and q^ the 
coefficient of induction. 

We take u and v as coordinates, given by the dipolar system 

x+yi = kt&u^(u+vi), (53) 

so that u = constant represents a circle through the poles 
(0, k), and v = constant represents an orthogonal circle, with 
the poles as limiting points. 

Now, if we revolve this system about the axis Oy, which 
may be supposed vertical, the two spheres, if outside each 
other, may be supposed defined by 

v = a and v= /3, 

so that a = fc cosech a, b = k cosech /3, c = /j(cotha + coth/3) ; 
and putting a + /3 = aJ, Maxwell shows, by Sir W. Thomson s 
method of successive images, that 

q aa = kZ cosech (n?3 /3), q a b = &2 cosech nft, 

q bb = k2 cosech(?iT a), (54) 

the summations extending for all positive integral values of n 
from 1 to oc . 

Here q ab is called Lambert s Series ; it is considered in the 
Fundamenta Nova, 66. 



288 THE RESOLUTION OF THE ELLIPTIC FUNCTIONS 

Again, with a ft = x, 

= k2 cosech 



and by the preceding formulas it can be shown that 

Trr/ 



................ (55) 

When the two spheres are equal, x = Q, and 
= = k% cosech Zn 



- 

When /3 = 0, the sphere ft becomes a plane; and now 
q aa = q ab = /c2 cosech Tia = a sinh aS cosech net ; 

which shows that the capacity of a sphere of radius a is raised 

from a to ct sinh aS cosech Tia by the presence of an uninsulated 

plane at a distance a cosh a from its centre. 

Similar functions occur in the determination of the motion 

of two cylinders or spheres, defined by v = a and ft, when 

the interspace is filled with homogeneous frictionless liquid. 
(W. M. Hicks, Phil Trans., 1880 ; Q. J. M. t XVII, XVIII. ; 

Basset, Hydrodynamics, I, Chaps. X., XL ; C. Neumann, 

Hydrodynamische Untersuchungen. ) 

270. To illustrate geometrically the singly infinite product 
forms in 263 of the elliptic functions, consider the analogous 
problems of electrodes at the corners of curvilinear rectangular 
plates, bounded by arcs of concentric circles and their radii. 

The vectors from the centre as origin of a series of p 
electrodes, equally spaced round a circle of radius a, will be 

aexp 2r?ri/p, where r = l, 2, 3, ..., p\ 

and with polar coordinates r, 0, the vector of the point will be 
T exp i6 ; so that for the p electrodes, each conducting a current 
of 2?r amperes, the vector function is 

loglfjr exp(i#)-a exp(2r7ri/p)} =log(r^e ? ^-a^), ..... (56) 



by De Moivre s Theorem (Hobson, Trigonometry, Chap. XIII.). 
Interpreted geometrically, the norm is the logarithm of the 
product of the distances of an} 7 point P from the electrodes, 
while the amplitude is the sum of the angles the lines joining 
the electrodes to P make with the vector $ 0. 



INTO FACTORS AND SERIES. 289 

We thus prove incidentally one of Cotes s theorems, namely. 
that the square of the product of these distances is 

( r p e ipe - a P)(rPe ~ W -aP) = r 2 ? - 2a*r*cos p6 + a 2 ^, ... (57) 
and, in addition, the theorem that the sum of the angles the 
vectors from the electrodes to P make with the vector $ = is 

r^sin pQ 
tan 1 ; ..................... (08) 

a p 



and when the sum of these angles is constant, the locus of P is 
an oblique trajectory of the curves 

rPcospO or r^sin pO = constant. 

With a single negative electrode at the centre, of current 
nw amperes, half the total current from the n electrodes on the 
circle will flow to 0, the other half flowing off to infinit} . 

Now the vector potential is, on writing e p for r/a, 

j. r n sin n 

i tan 1 



- 
r n cosnO a n 

We can isolate a sector, bounded by = 0, 9 = ir/n t and 
/ = a; and the preceding expression will represent the vector 
function of the electrical flow of JTT amperes, with electrodes 
at the end of the vectors r = a, and at r = 0. 

The amplitude of this expression will also represent the 
temperature in this sector, if the radius 6 = is maintained at 
temperature 0, while the radius (9 = 7r/?i and the arc r = a are 
maintained at temperature |TT. 

271. Now suppose that on the same circle r = a, an equal 
number p of negative electrodes are placed, equally spaced be 
tween the positive electrodes ; the vectors of these electrodes 
being a exp(2?^ l^i/p, the vector function is 



or, if moved out radially on to a circle of radius b, 

-log^-PeW + bP) ......................... (60, 

The vector function of p equal electrodes at a exp ZTTTI/}), 
and of p equal negative electrodes at a exp(2r l)-7ri/p will 
therefore be log^e?* 6 aP)/(rPe ip6 + a*) ; 

which, when resolved into its norm and amplitude, is 



--H tan~ : 
- 7-* -t-za r* cosptJ-|-a** 

J.E.F. T 



290 THE RESOLUTION OF THE ELLIPTIC FUNCTIONS 

, . cospO , . , , sin pO 
= tanh 1 r^- 4^ tan" 1 . .......... (61) 



with p = log(r/a) ; this function will represent the state of 
electrical motion in a wedge bounded by $ = and Q = Trjp. 

272. The substitution in the preceding expressions in 24-7 
of the conjugate functions pO and log(r/a)^ or pp for u and v, 
leads to the solution of corresponding problems for curvilinear 
rectangles bounded by arcs of concentric circles and their radii ; 
and now q=(b/a) p , where a and b are the radii of the curved 
sides, while tr/p is the angle between the straight radial sides ; 
so that in the rectangle OABC, 

OA = a7r/p, BC=b-7r/p, OC=AB = a-b. 

The vectors of the imaes of an electrode at are now 



where n denotes any integer, positive or negative, and 

r = l, 2, 3, ...,n. 
For electrodes at A, B, C, the vectors of the images are 



For a given value of n, the vector potential of the electrodes, 
whose vectors on a circle of radius aq nfp are 

aq^Pexp^riTr/p or aq nlp exp(2r ty-jri/p 
will be \ogU(r^ d -aPq n ) or logU(r p e ipd + a p q n ) .......... (62) 

Now, suppose a positive electrode is placed at and a 
negative electrode at C, with the corresponding system of 
images ; the vector function is 

low "i 



on introducing a negative electrode, of current TT amperes, at 
the origin ; and, writing irW/K for p6 + ilog(a/r)P, this becomes 



INTO FACTORS AND SERIES. 291 

equivalent, as in 263, on omitting constant terms, to 
log sn iv. 

A similar procedure with electrodes at A, C, and B, C, will 
lead to the singly infinite factorial expressions for en it, and dnu. 

Projecting these equipotential and stream lines stereographi- 
cally on a sphere which touches the plane, we shall obtain the 
corresponding solutions for the flow of electricity on the surface 
of the sphere. 

(Robertson Smith, Proc. R. S. of Edinburgh, vol. VII. ; 
M. J. M. Hill and A. J. C. Allen, Q. J. M. y XVI, XVII.) 

273. When these electrodes are replaced by straight parallel 
vortices, perpendicular to the plane, which is taken as hori 
zontal, the potential and stream functions are interchanged. 

Suppose a vortex is placed at a point P in the rectangle 
OABC ; to introduce the restriction that there is no flow across 
the sides of the rectangle, we must suppose the motion due to 
vortices which are the optical reflexions of the point P in the 
sides of the rectangle ; the sign of the vortex being positive or 
negative according as the corresponding image has been formed 
by an even or odd number of reflexions. 

The vectors of the positive images will therefore be 



and of the negative images 

2ma + 2nbi z ; 
wh ere z = x -+- y i- , .- = x yi. 

The resultant current and velocity function at =+>;* will 
therefore be the norm and amplitude of 



_ 

( 2ma + 2nbi + -z )(2ma + 2nbi + {;+z )" 
At the point P, this vector function, due to all the other 

images, is therefore 



( 2 ma -f 2nbi + z z 

and writing r,- = -> an d 2l^ 

K a a b 

this may, according to 263, be replaced by 



292 THE RESOLUTION OF THE ELLIPTIC FUNCTIONS 

The stream function at P is therefore, disregarding constants, 

9% H 2 w- H% 2 w , v 



= | log(ns 2 K, ns 2 m) 

= J Iog{ns 2 (u, /c) + ns 2 0, * )-!} : ...(66) 
so that the curve described by the vortex is given by 

ns 2 (2 Kx/a, K ) + u/(2K y/b, K) = constant, ......... (67) 

and all the other image vortices keep up a symmetrical dance, 
by describing similar curves. 

274. The vortex is stationary when at the centre of the 
rectangle; and now, changing to the centre as origin, the 
vectors of the images are ma+nbi, where m + 7i is even for 
the positive, and odd for the negative images; so that the 
vector function of the motion is given by 



, n ,, cnw //% . v 

= log =Jlog T . ................. (68) 

en kw 1 + cnit; 

Expressed as norm and amplitude, as in 247, this function 
_ i I 1 en w 1 en w l 1 en w 1 -f en w 

4 cn^// 



,, cuvi cuu ,, snu dn vi 
= ^ log ---- r- - + 1 log - 
& 



. , . . 

en vi + en u sn u dn vi + dn u sn w 

, , en u , , sn u dn t i 

tanh ~ J . tanh ~ 



. - T - . 

en w dn u sn w 

. 4 .snttdn t //<r ., 

en v) + ^ tan -1 , ........... (09) 

dn u sn ?? 

with u ZKxja, v = 2K y/b; the modulus of the elliptic func 
tions of v being /. 

The equation of a stream line of liquid is therefore given by 

en u en v = constant, or 
cn(2Kx/a, K )cu(2K y/b, K) = constant ............. (70) 

Close up to a vortex the velocity according to these ex 
pressions would become infinitely great, which is physically 
impossible; but a solid core may be substituted for this central 
portion, and the shape of this core has been investigated by 
J. H. Michell, Phil. Trans., 1890. 



INTO FACTORS AND SERIES. 



293 



275. When a point is placed inside an equilateral triangle, 
the Kaleidoscopic series of positive images is given by the 
vectors z, wz, w 2 z, where z = x + yi, and w is an imaginary cube 
root of unity ; the negative images being given by z, coz , o)V, 
where z = x + yi ; the origin being at a corner of the triangle, 
and the axis of x perpendicular to the opposite side (Fig. 27, i.). 




(i.) Fig. 27. (ii.) 

In addition, similar groups of six images must be added, 
ranged round the centre of hexagons forming a tesselated pave 
ment, the vectors of the centres of the hexagons being 
2mh + 2nhij3 and (2m + l)/i + (2?i + 1)^^/3, 
where h denotes the altitude of the equilateral triangle. 

In the corresponding doubly inh nite products, the elliptic func 
tions will have K jK= l J% i so that ( 47), /c = sin 15, 2or = J. 
Then, in Weierstrass s notation, the vector potential at 



for a single source or electrode inside the triangle will, neglect 
ing constant terms and factors, be expressed by ( 278) 
lo g o- (-s V (f-fttf )(r (i-orz) 



(71) 



while for a vortex or electrified wire, the vector potential is 



The nature of the resolution of these functions into their 
norm and amplitude is illustrated in 227 to 231. 

(O. J. Lodge, Phil. Mug., 1876; 0. Zimmermann, Das logar- 
ithmische Potential einer gleichseitig dreieckigen Platte, Diss. 
Jena, 1880 ; A. E. H. Love, Vortex Motion in Certain Triangles, 
Am. J. M., XI.) 



294 THE RESOLUTION OF THE ELLIPTIC FUNCTIONS 

So also for a rectangular boundary OACB, if we write 
a for g-x + (r]-y)i, or -2, 
ft for g+ x + (i-y)i, or +/, 
y for +a + (>? + 2/)i or f+z, 
5 for ^-a + fo + Z/)*, or faf; 

3;, -2 , 0, 5? being the vectors of the point P and its images 
by reflexion in the coordinate axes Ox, Oy, taken in order in 
the four quadrants ; then the vectors of all the other images 
by reflexion in the sides of the rectangle OABC being ranged 
in a similar manner round points whose vectors are 2ma-f 2?i6i, 
it follows from what has gone before that we may express the 
vector function at f of all their images, taken as positive, by 

log era v/3 o-y rS, ........................ (73) 

with w l = a, a) 3 = bi ; 

disregarding constant factors, and exponential factors of the 

form exp(Au + Bu 2 ). 

But when we represent the vector potential of a vortex or 
electrified wire at P, the vector potential becomes 



276. As another illustration of the connexion of a regular 
Kaleidoscopic figure with Elliptic Functions, consider the solu 
tion of the reciprocant 

8 = 0, .................. (75) 



dii d 2 ii 7 d s y d*y 

where ^ 8B 7 i a = j v ^="T^> c= i 

dx dor dor dx* 

(Sylvester, Lectures on the Theory of Reciprocants, VI., 1888.) 
Mr. J. Hammond has shown (Nature, Jan. 7, 1886, p. 231 ; 
PTGC. L. M. S., XVII., p. 128) that the integral of this equa 
tion (75) may be written 

(l + ti)dt ( n 

> 



By turning the axes through an angle ^tan^A/ic), we can 
make X vanish ; and now, replacing JAC by unity, 



^-^5 0,4),. ..(78) 



and K* + y*)K-2^)= 1 ......................... ( 79 > 



INTO FACTORS AND SERIES. 295 

Since (196) $>(joz = copz, $>a) 2 z = a) z pz, 
where eo is an imaginary cube root of unity, therefore 

pu(x+yi)&u?(x-yi) = i, .................... (80) 

which shows that the curve is unchanged if turned through an 
angle of 60 about the origin (Fig. 27, ii.). 

Captain MacMahon has shown that the intrinsic equation of 
this curve may be written 

cos3^ = cln(s/c), with jc=JV 2 ................ ( 81 ) 

The student may also show that the equation of the curve 
may be written in one of the forms 



K 2 sn 2 (#, K) = ic /2 sn 2 (y, AC ), 

K)dn(y, K ) = K, .............................. (82) 

with /c = sinl5, /c = sin75. 

As a similar exercise, the student may solve the rcciprocant 
fc-56 = .............................. (83) 

in the form $>x $>y = -1, ............................ (84) 

and determine its intrinsic equation, drawing the correspond 
ing curves (Proc. London Math. Soc., XVII., p. 360). 

277. When we expand, in ascending powers of u, the 
logarithm of a doubly infinite product, such as that in the 
numerator of sn u in equation (1), 258, we find 



Now, when the origin is taken at the centre of all the 
points whose vectors are Q, the coefficients of u, u 3 , u 5 , ... 
vanish ; but the value of the series is still indeterminate, until 
the infinite curve containing all these points has been defined. 

For if P denotes this infinite product, and P its value when 
the boundary has changed into a similar curve, then 



where the summation now extends over the region lying be 
tween the two boundaries; and now the limit of SQ~ 2 is a 
definite number, A suppose, while the limit of 2Q~ 4 . ... is zero. 

Therefore 

logP / -logP=.Wu 2 , or P = Pe* Au *...-, ......... (86) 

so that the value of the infinite product depends on the shape 
of the infinite boundary (Clifford, Math. Papers, p. 463). 



296 THE RESOLUTION OF THE ELLIPTIC FUNCTIONS 

But, as in $ 261, Weierstrass removes this ambiguity \yy 
attaching to each linear factor of the product, such as 



,. i f (u 1 

an exponential factor exp( c -f - 

\l2 z 

and, in the physical analogue, the corresponding electrode at 12, 
whose vector function is log(l u/Q), must have associated 
with it a uniform flow in the direction of the vector Q, repre 
sented by u/Q ; and a streaming motion iri rectangular hyper 
bolas, whose asymptotes are parallel and perpendicular to the 
vector Q, represented by |(u/Q) 2 . 

Now in the expansion of the logarithm of the doubly infinite 
product P, when these exponential factors are introduced, 

logP = logu-XZQ- 4 -iu 6 Sft- 6 -..., ........... (87) 

an absolutely convergent series ; that is, a series the value of 
which is independent of the order of the terms. 

278. Making a new start ab initio with the sigma func 
tion ( 195), as defined now by the equation 



n/ TT/ It > - /TT\ 

n l-exp+5 , ......... (U) 



where Q = 2mto + 2iico , and ta /wi is a real positive quantity, so 
that co, a) correspond to w> v co 3 or co 2 , a) 2 according as A is posi 
tive or negative, then a-u is the analogue of Jacobi s Eta Func 
tion ; i n fact, 

a-u = Ce Au R^(e l - e s )u = Ce^O^TrU/u), ....... (88) 

( 263), where C, A are certain constants ; also log a-u is the 
same as logP in equation (87). 
Now denoting, as in 195, 



d log o-u , - , cZ 2 lo(r aril du 

~ by ^ and - or 



(V) 



by differentiation of ( U) and (58) ; so that, on reference to 195, 

we may put 

# 2 = 602Q- 4 , # 3 = 14.0ZQ- 6 , ............... (W) 

also 2 = 2 4 .3.5 2 .72Q- 8 , = 2 4 . 3. 5 . 7. 11 2Q 10 , etc. 



INTO FACTORS AND SERIES. 297 

Differentiating (60) again, 



p tt= -- r (,7Q? ......................... (Y) 

Then (o-u)/u, ufu, u 2 ^u, u 3 ^ u, u*p"u, ..., are unaffected by 

the considerations of homogeneity of *196; as for instance in 

the expansions in equations (21) and (22) on p. 249. 

A change in (X) and (Y) of u into u + 2pw + 2qw, where p and 

q are integers, merely leads to a rearrangement of terms ; so 

that, as in 250, 

p( U + Ipw + 2^0) ) = pit. 

Also, since in Q = 2m&>-f2?7a> , the arrangements (771, -?i) and 
( ?7i, ri) exist in pairs, therefore 



and p /2 u = 4 . pw, p w . pu, 

= lp*u-g 2 pu-g 3 , ................................. (AA) 

as originally defined otherwise in 50. 

A change of u into u + 2w in (V) shows that, by a rearrange 
ment of terms, 

ftw + 2ft>) = 6 + 217, ..................... (89) 

where q is a certain constant, determined by putting w= w, 
so that / = & ............................ (90) 

Similarly ^ + 2o/) = fu + 2i/, ..................... (91) 

where ? $ ; .......................... (92) 

and, generally, 

f(u + 22^ + 2go) / ) = t + 2p, ; +2^ / ............ (BB) 

Integrating (^9) and (90), 

<r(u + 2w) = Ce*i<ru,, <r(w + 2o) ) = WVw ; 

where (7 and (7 are determined by putting u = oo and a/ ; 
so that 

(7(u + 2o))= _ e *+)(rM, o-( + iV)= -e*^ M + w Vit, (93) 
and therefore 

,, ..................... (94) 

, ..................... (95) 

and, generally, 

(f(u + 2^0) + ^gco ) = - ( - 1 )(P+lX?+l) e C2,^+2 9 7 7 Xt 4 +; )w + 9 a; ) (rM/) . . . (CO) 

obtained also by integration of (BB). 



298 TH E RESOLUTION OF THE ELLIPTIC FUNCTIONS 

The doubly infinite products in (U) may be converted into 
singly infinite products ; and now 



where q = e l", and 



trie* = ix 2 - 7r 2 S , j = J-TT* - 7r 2 2 cosech 2 (W/a>;), ... .(97) 

etc. ; for the proof of these and other similar formulas merely 
stated here, the reader is referred to Schwarz and Halphen. 

Also, denoting Q + w, ft + + , Q + w by Q x , fi 2 , Q 3 , 
then the function cr a % of 202 may be otherwise defined ab 
initio by the relation 

^ ^ = ^Iin(l-)exp(J + Ig), ........ (EE) 

which will be found to lead to the preceding results. 

d 2 
Denoting j- 2 logout* by <p a u, we shall find that 

&y* = K + a>a)> = 1 2 > 3 .................. ( 98 > 

(A. R. Forsyth, Q. J. M., XXII.) 

279. Returning to the function G of equations (8) and (10), 
215, and changing the sign of the us, we may also write it 
_ cr(v + Ui + U 2 + + UH)<T(V ~ UI)<T(V U 2 ) - . . <r(v u u ) 



(99) 

and since we may suppose the it s and v to be all increased by 
equal amounts, the condition (9) of 215 is no longer required. 
Now, since G vanishes when v = u r , where r=l, 2, 3, ..., /x; 
therefore the coefficients c , c v c 2 , ..., C M are determined by 
a series of equations of the form 

= c o + c i^^ + c 2F / u r+...+c^- 1 )it r ; ......... (100) 

and therefore the determinant 

(101) 



where M is a factor independent of v ; and now this theorem, 
as a corollary of Abel s theorem, shows that the determinant 
also vanishes when v= u-^ u . . . u^. 



INTO FACTORS AND SERIES. 290 

The symmetry of the determinant shows that M must be a 
symmetric function of the us ; or writing u for v, and denot 
ing the determinant by <j>(u Q , u lt u. 2 , . . . , Up\ then cp is a 
symmetric function of the it s, such that 

. x _ A 

u . . . i - 



and it will be found (Schwarz, 14) that 

J. =(-1)^-1)1! 2! 3! .../il. 

Thus, for instance, with JUL = 2, 
1, pw, p zi, =2 
1, pv ^ 

By forming a similar function C" of the u"s, subject to the 
condition (6) of 215, we see that (7) is an elliptic function of 
v, which can be expressed by C/C , where C and C f are given 
by determinants, as above. 

Equation (CC) is also sufficient to prove that the function 
in (7) 215 is doubly periodic. 

As an application of the principles of this article and of 
209, 215, 216, 257, the student may prove that Q of 215 is, 
writing a for u v b for u.- and u for v, given by the equations 

)o-(u + b)ar(a + b) 



<r(u + a -f- b)a-u a-a orb 



1 , pa, 



1, pa, p a 



1, p6, p ft 

We thus verify the equations of 209, 257, 
du 



= $(u, a)<t(u- t b). 

When condition (6) of 215 is not satisfied, then (7) reappears 
qualified by an exponential factor of the form e pv when v is 
increased by 2>o) + 2go/; the function is then called by Hermite 
a doubly periodic function of the second kind ; the function 
(p(u, v) defined in 201 being the simplest instance of this 
kind of function. 



300 THE RESOLUTION OF THE ELLIPTIC FUNCTIONS 

280. Making the i/, s all equal, as in 218, and interchanging 
u and v, the function 

rfu+ftf 



is a doubly periodic function which can be expressed in the 
form of C ; but now the coefficients c must be determined by 
a series of equations of the form 



Expressed as a determinant we may now put 



pu-pv, 



Finally, making u = v, and dividing both sides by 
we find, in the limit, 

l=M * "" 



where 



Halphen denotes this function of u by ^(p+ 
Thus for instance, as in 200, with /* = 1, 



= " ~2 (Schwarz, 15); 



. t =-pu. 



Again, with /x = 2, 



By logarithmic differentiation, 









lo ^n^ = -- lo , = 7i 2 ^ - 



whence $nu can be expressed rationally in terms of 
When u = v, 



, . . . .(HH) 
, $> u, ____ 



INTO FACTORS AND SERIES. 3Q1 

Also, when u = 0, 



= a M (-l>* + V ; ............................ (102) 

and therefore a^ = 0, when jj.v = Sp^ + 2<?o> 3 . 
281. In the pseudo-elliptic integrals ( 218) 

yuv = (mod. w v o> 3 ); 

and now, knowing the number /z, the coefficients c , c v &,, ... in 
C or xu are readily calculated from a knowledge of the values 
of py, p v, p"v, ... ; in this way the results employed in 218, 
219, 223, 225, 233 were inferred. 

Thus, for instance, in 219, we know that 

JUL = 3, JULV = 3a) x + o> 3 ; 

pv = J, p v = 3V 2 > P^ t= -6, p" v = 18i^/2, >"% = -252, ...; 
so that the ratios of c , c p c 2 , . . . can be calculated from the 
equations = c + ^4- 3i^/2c 2 6c 3 , 



0- -6^ + 18^2^- 252c 3 . 

Taking an arbitrary value of c 3 , say f, we find, by solution, 
V=-9, c 1= -10, c 2 =-3V2; 
^ = f c s (f p"u 31^2 p w, 10 j?u 9) 
= fc 3 {(2 p 

Now u 



so that, in the algebraical herpolhode referred to axes rotating 
with a certain angular velocity, we may put 



thus leading to the results of 219. 

As other numerical examples the student may investigate 
the results of 218, 223, 225, 233 ; also the example due to 
Abel (CEuvres, I, p. 142), where yu = 5, # 2 =12, # 3 = 19, and 
v = f w 2 or ift> 2 , when <@v = 2 or 1 ; we then find that the 
values of c , c v c 2 , c 3 , c 4 , c. are proportional to 

-288, -36, -48^3, 12, ij 3, 0; 
or -396, -252, -12i^/3, -24, ^3, 0. 



302 THE RESOLUTION OF THE ELLIPTIC FUNCTIONS 

Writing s for $u, then we may put 
X u = - 288 - 36#m - 48i^/3p ie, + 1 2%>"u + i J 3p"u 

= 36(2s 2 -s- 10) + 12i x /3(s-4) /v /(4.s 3 -12s- 10), 
M = - 396 - 252w- 



We thence infer that the corresponding pseudo-elliptic inte 
grals involve 



_ 1 ._ . 



2(8-1) 



. tan -_ > 
~ 



and now by differentiation we infer that 



2s + 13 cs_ 2_ _ ,(8 -4)^/^-1^-1 9) 

!^" 

s 



__ 
" 



* ~ /" r> " x-*. ~t y-v \ "" /-A Ldll 

S 1 

J s + 2 

Thus, in the Weierstrassian notation, 

= - tan ~ 

V)W <0V 






with f/ 2 = 1 2, # 3 = 1 9, according as yv = 1 or - 2. 

These results may be employed in the construction of 
degenerate cases of the catenaries discussed in 80, 205, 206. 

Thus, for instance, the curve given by 



r ^o^( 2^/3u-oO~) = ^/^ 
is a plane catenary for a central attraction ri*wr per unit of 
length, in which ( 80) 

So also a tortuous catenary is given by the equations 



/ / 5 cos(5$ + 
under an attraction nhvr to the axis Ox. 



INTO FACTORS AND SERIES. 303 

282. Other pseudo-elliptic integrals are formed by the sum 
of two or more elliptic integrals of the third kind, when the 
sum of the parameters is of the form pw + qco, as in 226, for 
the expressions of and . 

We shall denote the integral of the third kind in the form 
(fa), 199, by $(u, v), as this we have found is the form of 
most frequent occurrence in the dynamical applications ; and 
now (fa) shows that 

$(u, a + b) 

,11 <r(a u)<r(b 
+Mog-7 i \ 
8 <r(a + u)o-(b + 

IQ 



pa p 
by reason of (y), 197, and (K), 200 

When a + 6 = a> a , p ( a + 6) = 0, $(tt, a + 6) = ; and now 



, 6)= - 



By equation (N), 249, we may write 

i lo<y g(a + .)- g q = tanh , 1 /Am-e a . pa-q, . pq- 
p p(a u) e a \ \pa ea.pa ep.jpu 

, , p a pu e . . / /a pu 

= tanh- J , , or % tan- 1 f 





pa e a 

the latter form to be employed in dynamical problems, where 
p a is always imaginary ; thence the expressions given for f 
and in 226 can be inferred. 

As an application we can put a + b = a) l + (a s or co 3 in 209, and 
thence deduce a degenerate case of the Spherical Pendulum. 

EXAMPLES. 
1. Prove the following q series : 



(i.) 

... 

... 90 



(iv.) (1- 

(v.) ^/(KK)^. 2q^, g^ T V/cV 2 , J"sl/l72S</ 2 , or l/1728g, accord 
ing as A is positive or negative, when q and K or // is small. 



304 THE RESOLUTION OF THE ELLIPTIC FUNCTIONS 
2. With the notation of g 265, prove the theorem 



= 2<9 1 <6> 1 (6- - y - z)0 l (s -z- x^s -x-y}, 
where 2s = w+x + y + z. 

Deduce the formulas 
(i.) A /2 sn u sn v sn r sn s 

Ac 2 cn u en v cri ? en s + dn u dn v dn r dn s K Z = 0, 
provided u + v -\-r-\- 8 = 0. 

(ii.) K 2 sn -J(u -}- v + r + s)sn | ( & + v r s) 

X sn ^(u v + r )sn J(w v r+s) 



3. Show that 



= 2(e 2 - e s ) (e s - ^(^ 

4. Show that Weierstrass function a(u) satisfies the partial 
differential equations 



Show that the second of these equations is also satisfied by 
the function 

cra(u)/{ (e a - ep)(e a - e 7 ) }i ; 
and write down the differential equation satisfied by o- a u. 

5. Prove that the projection of a geodesic on a quadric of 
revolution on a plane perpendicular to the axis is analytically 
similar to a herpolhode (Halphen, II., Chap. VI.). 

6. Evaluate the surface of an ellipsoid. 

7. Construct some degenerate cases of trajectories or caten 
aries on a sphere, or on a vertical paraboloid or cone, employing 
the numerical results of the pseudo elliptic integrals. 



CHAPTER X. 

THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 

283. By the Theory of Transformation is meant the ex 
pression, in terms of the elliptic functions of modulus K and 
argument u, of an elliptic function with respect to a new 
modulus X and of a proportional argument u/M; and then M is 
called the multiplier, and the relation connecting the moduli 
X and AC is called the modular equation. 

A particular case of Transformation has already been intro 
duced in Landen s Transformation ( 28, 67, 71, 123, 181, 182) 
in its application to Pendulum Motion, and to the Rectification 
of the Hyperbola. 

In accordance with the plan of this treatise, we begin with 
a physical application of the Theory of Transformation, before 
proceeding to the analytical treatment of the subject. 

Suppose then in 259 that an odd number, n, of such 
rectangles as OABC are placed in contact, side by side, so as 
to form a single rectangle OA n B n C, of length OA n = na, [and 
height 0(7=6 ; and now put 



OA /OC= a/5 = A/A , 
so that A /A = nK IK , ........................... (1) 

where K, K denote the quarter periods with respect to the 
modulus K ( 11), and A, A with respect to the modulus X. 

Let us begin by placing a positive electrode at 0, and an 
equal negative electrode at (7; then, inside the rectangle OB, 
the vector function will be 

log sn Az/a = log sn(Ax/a+A iy/b), 
with z = x + yi. 



G.E.F. 



306 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 

But, inside the rectangle OB n , the vector function of these 
electrodes and their images will be that due to positive elec 
trodes at 2sa and negative electrodes at 2sa+bi, where s 
assumes all integral values from to n 1; and the vector 
function of this system is ( 259, 275) 

s = n-l 

log II sn K(z-2sa)/na = log ILsu(Kx/na+K iy/b- 2sK/ri). 

s = Q 

The physical equivalence of these two forms of the vector 
function, as seen from two different points of view, shows that 



or sn(u/lf, X) = A li sn(u 2sK/n), (2) 

where u/M= Az/a, u = Kzjna ; 

so that M = K/nA = K /A ; (3) 

this is the formula for HIQ first real transformation of the sn 
function, of the nth order. 

Similar considerations will show that 

>, (4) 

(5) 

If, as in 263, we put 

q = exp( TrK jK), and r = exp( TrA /A) ; 

then r = q n , (6) 

and X is less than /c. 

It simplifies matters to place the rectangle OB in the 
middle of n such rectangles placed side by side, and now s 
ranges from ^(n 1) to J(?i+l); and combining equal posi 
tive and negative values of s, we find, according to (7) 137, 



=!(- 1) sn 2 ?y, sn 2 2s 

TT OiJL W/ O1JL o(jU 

H 



s= i JL rr 

where co = K/n ; 

oi> y=^ n i 1 rXv y (8) 

JjfJ. .L ^^ /C (JL w 

connecting y = sn(u/M,\) and x = su(u, /c), a = sn(2sK/n). 

284. Next suppose that w equal rectangles, such as OABC, 
are piled on each other, so as to form a single rectangle 
OAB n C n , where OA =a, OC n = nb ; and now put 



THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 307 



OA/OC = a/6 = A/A ; 
so that K IK=nA!/A ..................................... (9) 

The physical equivalence of a positive electrode at and an 
equal negative electrode at C, and of their images in the rect 
angle OABC, with the positive electrodes at 2sK iy/b and the 
negative electrodes at (2s+l)K iy/b in the rectangle OAB n C n 
and their images, shows in a similar manner that 

sn( Az/a, X) = A II sn(Kx/a + K iy/nb - ZsK ijn), 
where s may assume all integral values from to n l, but 
preferably, from J(??, 1) to i(?i + l); or 

sn(u/M, X) = A II sn(tt - ZsK i/n, K ), ........... (10) 

where u/M Az/a, u = Kzja ; 

so that M=K/A = K /nA -, ..................... (11) 

and now, with 

q = exp( TrK jK), r=exp( xA /A), 
we have r = q 1 / 1 , .............................. (12) 

and now X is greater than /c. 

Similar considerations show that, by placing positive and 
negative electrodes at A and C, or B and C, we shall obtain 
the formulas 

cu(u/M, \) = BTL cu(u - 2sK i/n) ; ............ (13) 

du(u/M, \) = CIL dn(u - ZsK i/n) ; ............ (14) 

these are the formulas for the second real transformation of 
the elliptic functions, of the Tith order. 

A similar physical interpretation of Transformation may be 
given in connexion with the curvilinear rectangles bounded by 
concentric circular arcs and their radii, as discussed in 270. 

285. Besides the first and second real transformations in 
which q is changed into q n and q lin , now denoted by r^ and 
T O , there are in addition n 1 imaginary transformations, 
when n is a prime number, in which q is changed into w p q l/n , 
denoted by r p , where p = 1, 2, 3, ..., n 1, and o> is an 
imaginary Tith root of unity ; so that, corresponding to a given 
value of K, the modular equation of the ?ith order, if prime 
will be of the (?i + l)th degree in X, having the roots 

^oo \) ^l ^2 ^n-l> 

of which two only, X and X , will be real ; \ x < K < X . 



308 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 

We need only consider the Transformations of prime order, 
as a Transformation of composite order, mn, can be made to 
depend on the transformations of the mth and nth order. 

The different transformations of the mrith order are formed 
by changing q into <? m/n ; so that the number of transformations 
for any number in general is the number of divisors of mn ; 
reducing to ?i + l, as before, for a prime number n. 

For a transformation of order ri 2 there is one real transforma 
tion for which q remains unaltered, and we thus obtain the 
formulas for Multiplication of the argument u by n. 

286. After this physical introduction, we can proceed to the 
general algebraical theory of Transformation, as developed by 
Jacobi in his Fundamenta nova theories functionum ellipti- 
carum, 1829. 

The theory in its generality consists in the determination of 
y as a rational algebraical function of x, of the form 

y=Uir, ............................. (15) 

where U and V are rational interal functions of x, 



so as to satisfy a differeotial relation of the form 

Mdy dx 



where X= ax* + 4bx*+ 6cx 2 + dx + e, \ 

Y=Ay* + 4,By s + 6Cy 2 + 4<Dy + E,)" 

Making the substitution of (15), we find that we must have 

dx 



and the first condition requisite is that 



where T is a rational integral function of x, of the (2n 2)th 
degree ; and now, if we can make 

(20) 



where If is a constant multiplier, the Transformation is 
effected. 



THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 309 

But if U and V are both of the nth degree, or if one of the 
nth and the other of the (n l)th degree, so that either a n or 
b n (not both) is zero, this is necessarily the case ; for any 
square factor in (U, F) 4 will appear as a linear factor of 

dU n dV 

j V U =j j 

dx dx 

which is also of the (2n *2)th degree, and can therefore only 
differ from I 7 by a constant factor M. 

The Transformation is now said to he of the nth order. 

By taking X of the sixth, instead of the fourth degree, Mr. 
W. Burnside has derived hyperelliptic integrals (Proc. L. M. S., 
XXIII.) from the elliptic element dy/*JY, similar to the hyper- 
elliptic integrals of 159, 160, by means of substitutions of 
the second, third, and higher orders. 

Now denoting by a, /3, y, S the roots of the quartic X = 0, 
and by a, /3 , y, S f those of Y ; so that, resolved into factors, 

X= a(x-a)(x-/3)(x-y)(x-S), 
Y=A(y- a!)(y - /3 )(y - y }(y - <T) ; 
then A(U-aV)(U-/3 V)(U- 7 V)(U-S V) 

= aT\x-a)(x-fi)(x-y)(x-$} , 

and now a factor, such as U a V, must be composed of linear 
factors, such as x a, and of the squares of factors of T. 

In the expression y = U/V there are at most 2n + l arbitrary 
constants ; and in determining 7 and Vso as to satisfy relation 
(19) we determine 2n 2 of these arbitrary constants; thus 
there remain at disposal three arbitrary constants, correspond 
ing to the three constants involved in an arbitrary linear 
transformation, such as that obtained by writing ( 139) 

(lx+m)l(l x + m } for x, 

as exemplified in 153, 160, where the constants I, m, I , m 
are chosen so as to make X and Y quadratic functions of y? 
and ?/ 2 . 

When X and Y reduce to quadratic functions of x and y, 
the elliptic functions degenerate into circular and hyperbolic 
functions : and now there is no Theory of Transformation, 
except for the change from circular to hyperbolic functions, as 
in 16. 



310 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 

287. Jacobi, in his Fundamenta nova, works throughout 
with the differential relation for the sn function ( 35) 

dx 



connecting x = su(u, K) and y = su(u/M, X). 

Now, if y=U/V, 

then, since u = makes x = and y = 0, y and therefore U 
must be an odd function of x, the other, V, being an even 
function ; so that for an odd order of the transformation 



Since x = I, y = l: X = !/K, ?/ = l/X; etc., are simultaneous 
values of x and y, the relation connecting x and y may be 
written in any one of the following forms, 

1+ y = (I+ x)A*/V, or V+ ff=(l+ x)A*; 

1_ y = (l- x)A /z /V t V- U=(l- x)A *; 



(I- K x)C 2 ; ..... (22) 
where A and G are rational integral functions of x, of the 
^(n l)th degree, which become changed into A and G when 
x is changed into x ; so that we may put 
A=P+Qx, A = P-Qx, 
C^P + Vx, C = P -Q x, 

where P, Q, P , Q are even functions of x ; and therefore 
l-y l^-ocP-Qx\* l-\y l^ 



1 + y l+oj\P+Qaj/ l + \y l+KX\P -Q x) 

O^l VinO* W ~"^ $/ *"~: - a ^ a ( ^O ) 

When the order n of transformation is even, we put 



and now 7+ U = (1 + x)( 1 + Kx)B*, V+\U = D 2 , 

y-^=(l_a;Xl-^)5 /2 , F-X^7=D 2 ; ......... (24) 

where 5, D are rational integral functions of x, of the (^n l)th 
degree, changing into B f and D when x is changed into #; 
so that we may put 

B = R+Sx, B = R-Sx- 



where R, S, R t S f are even functions of x. 



THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 31 1 

288. The number of independent constants represented by 
the a s and 6 s in U and V can be immediately halved by 
noticing that a change of u into u + K i has the effect of 
changing x into l/ K x and y into l/\y ( 239); and therefore of 
interchanging U and V. 

An algebraical simplification is thus introduced by writing 
X /\/K for x and y/^/\ for y, as in 143 ; the differential rela 
tion now becomes of the form (Cayley, American Journal of 
Mathematics, vol. 9) 



P dx 



(25) 



and 2a = jc+l/K, 2/3 = X + l/X, .................. (26) 

, . sn(u, /c) sn(pu, X) . 

connecting x =, -, y *-~ - 

v K 
and now, if y= U/V, 



for an odd order n of transformation, involving only n co 
efficients B , B lt ... , B n _i, and therefore n l arbitrary 
constants in y ; also B n _i=pB . 

It follows then that, in the original relation y=U/V, con 
necting aj = sn(w, K) and y = su(u/M,\), if a 2 x 2 is a factor 
of 7, then 1 /c 2 a 2 x 2 must be a corresponding factor of V ; and 
we thus obtain the expression of y as a function of x given in 
equation (8), and in addition the relation 

\ = M*K n ILa*, (27) 

so that we may write 

y = M^xILj^_~^ (28) 

Professor Cayley writes equation (25) in the form 

(i+ 



where the ^ s and S s are the zonal harmonics of a and /3. 
289. Writing this equation (28) in the form 



which is an equation of the ?ith degree in x, the roots of which 
are x = snu } sn(u2a>), ... , sn{u(?i 



312 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 

where o> = 2K/n or 2K i/n for the two real transformations, we 
find that the sum of the roots 



_ 

or combining the equal positive and negative values of s, 

\ ( u ^ \ 2 sn u en 2so> dn 

-^r^sm ^, X ) = sn u -h 2 = - 5 -: - 5 
/elf \lf / 1 /c 2 sn 2 2so>sn 2 u 



the expression for ?/ when the product in equation (8) is resolved 
into its partial fractions ; and similar expressions hold for the 
en and dn functions (Jacobi, Werke, I., p. 429 ; Cayley, Elliptic 
Functions, p. 256). 

290. We need not therefore confine ourselves, with Jacobi, 
to the Transformations of the sn function ; but we may some 
times find it preferable to seek the relations connecting 

x = cn(u, K) and y = cu(u/M, X), 
when ( 35 ; Abel, (Euvres, I., p. 363) 

Mdy _ dx 

/(! - 2/ 2 V 

or the relations connecting 

x = dn(u, K) and y = 



_-, 

2 ~ - 2 2 ~ 



relations already given in (4), (5), (13), (14) of 282, 284. 

But Prof. Klein points out (Math. Ann., XIV., p. 116) that 
it is the differential form of 38 (really Rieinann s form), 
connecting z = sn 2 (u, /c) and t = sn 2 (u/lf, X), 
and leading to the relation, on writing k for /c 2 and I for X 2 , 

Mdt dz^ _ _ , ,oo\ 

~ 



which is the most fundamental in the theory of the elliptic 
functions sn, en, and dn ; the periods now being 2/f and 2K i, 
instead of 4>K and ZK i, etc. ( 239) ; the quadric transforma 
tions (of the second order) 

z = x 2 , 1 x 2 , or 1 A 2 , 

t = y\ l-if, or 1-X 2 /, ................. (34) 

leading immediately to the preceding transformations of the 
sn, en, and dn functions. 



THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 313 

291. The Theory of Transformation may be developed en 
tirely from the algebraical point of view ; but Abel has shown 
how the form of the transformation of the nth order may be 
inferred from the elliptic functions of the nth parts of the 
periods, called by Klein, modular functions. 

Thus taking the first real transformation connecting 

in relation (33), then 

- -nd-*y -HA 



D= II(l-A;a0) 2 , ...................... (35) 

where a = sn*2sK/n, /3 = sv-(2s-l)K/n, 

arid the products extend for all integral values of s from 1 to 

!(-!> 

The form of the factors is inferred by Abel from the con 
sideration that 



(i.) 
where s and s are integers ; and, from equation (3), 



z = sn 2 2sK/n = Q, or a; 
(ii.) when = l, ^Y=(2s-l)A+2s A% 

u = (2s - l)K/n + 28 jfiT *, 

z = sn\2s- V)Kjn = 3 or 1; 

(iii.) when t = l/l t u/M=(2s-I)A + (2s -l)A i, 

u = (2s - 1) K/n + (2s - I)K i, 

z = sn*{(2s-I)K/n-K i} = l/k/3 or l/k. 

(iv.) when t = co , u/M= 2s A + (2s - 1) A i, 



z = sti 2 (2sK/n - K i) = 1/ka, or oo . 
Similarly the relations can be inferred connecting 

3=cn 2 (tt, K ) and t = cn 2 (u/M, X), 
or z = du 2 (u, K) and t = cn 2 (u/M, X), 

not only for the first real transformation, depending on equa 
tion (3), but also for the second real transformation, depending 
on equation (11), and also for any one of the imaginary 
transformations of the nth order. 



314 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 
292. In Weierstrass s form the relation is 

dx 



connecting x = p(u; g 2 , g 3 ) and y = p(u/M; 72,73), 
by a relation of the form 

y=U/V; 
and this must be equivalent to relations of the form 

y-e a = (x-e a )A*/V, or (x-efi&IV, or (x-eJC*IV, (36) 
for a transformation of odd order; giving 



so that V must be a perfect square ; thus leading to the 
requisite number of equations for the determination of the 
arbitrary coefficients in U and V, and an equation over, which 
relation may be made to connect the absolute invariants J 
and J , and corresponds to the modular equation. 
For a transformation of even order, we shall have 

U 



equivalent to relations of the form 

9-+ 

and therefore 



A 2 x-68 B* x-e y <7 2 

si* r 5= 3* or ** ..... ( > 



293. In the Weierstrassian form we determine the relation 
connecting x = $>(u, J) and y=p(u/M t J ). 
But without altering J we may write ( 196) 



and now, if w, M denote the real and imaginary half periods of 
$>(&, J) or pu, we may take w/n, u> as the periods of ip(u, J ) in 
the first real transformation of the nih order ; and w, af/n as 
the periods in the second real transformation (Felix Muller, De 
transformations functionum ellipticarum; Berlin, 1867). 

The first real transformation, of odd order n t may now be 
written 



similar to equation (30) for the sn function, and obtained in a 
similar manner. 



THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 315 
By integration of this equation ( 195) 

S ~~ n), (41) 



8 = 1 

where G^l* 2 p(28a>/ri) = p(2sw/n) ; ................. (42) 

S=l 3=1 

and integrating again, 

log <r(u, J ) = (j^u 2 + log o-u II o-(u 2sa)/ri)(r(u + 2sco/ri), 

<r(u t J ) = Ce G ^o-u n <r(tt - 2sw/n)o-(u + 2so/%).j ............ (43) 

The constant C is determined by putting u = 0, when 



cru a-(u 

= n x 

(r(-2sa)/n)(r( 2sw/n) 
and now 

,r(, JQ = e^Wn" )(7(28 ^-:^ 2 r /n+M) 

a=l (7-(2Sft)/7l) 

p2sa>/7i), ...................... (44) 



by formula (K) of 200. 

Thus, for instance, with 7i = 3, 

o.(u, JO = e G ^(cm)*(& U - GJ, ............... (45) 

where (7 1 = p|o) = ^Jw, 

and therefore satisfies the equation of 149 






v ............... ( 46 > 

Denoting by 6r 2 and 6^3 the transformed values of # 2 and g B , 
they are found by a comparison of coefficients in the expansion 
of both sides of equation (44) in ascending powers of u ( 195). 

Thus, if J"=0, or g. 2 = Q, then 6^ = or ^ 3 ; and taking the 
value G l = 0, then J = 0, G 2 = Q, G B = -27g B , and 

<r(u; 0, -27g B ) = (<ruf&u ...................... (47) 

Employing the principle of Homogeneity of 196, this 
equation may be written 

(r(uiJ3) = iJ3((ru) 3 &u, ............... (48) 

leading by differentiation to 

itu> .............. ( 49 > 



and 3 P ("V3)= -3^+-= ~^ + ...... (50) 

since g 2 = 0, as in 47. 



316 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 



Thus, if </ 3 is positive, and <o 2 , o> 2 the real and imaginary 
half periods ( 62), then ft) 2 /a> 2 = i^/3 ; and if we take u = fo> 2 , 
then & 3 u = g 3 ( 166, 233) ; so that pw 2 / = 0. 

Again, putting u co 2 in equation (49) gives 

%V3 = 3jfe ........................... (51) 

Making use of the last equation of 202, we find 



As a numerical exercise the student may construct the 
following table, and also fill in the values for u = w 2 > o> 2 , Jo) 2 , 
-Jo> 2 , |-w 2 , |eo 2 , ... ; taking # 2 = 0, <7 3 = 1 ; these numerical results 
are useful in the problem of the Trajectory for the Cubic Law 
of Resistance, discussed in 227-234. 



,+ 



(V2+D 3 



7ry.3 

IW */2 + l) 3 e T "^ 



1 ^V3 
-* 3 

ie 



Linear Transformation. 
294. In Chapter II. the general elliptic differential 
has been reduced to Legeridre s standard form 



and to Jacobi s, or rather Riemann s standard form (11) of 38, 



by various substitutions, in 39, 40, 41, 42, 43, etc., which are 
practical illustrations of the Linear Transformation. 

In 160, the six linear transformations are given which, 
according to Mr. R. Russell, reduce 

dx/^/X to the form dz/J(Atf + 6Cz*+E). 
In determining the linear transformations, of the form 

y= U/r=(aSI> + p)l(yX + 8) ................... (52) 

which satisfy Riemann s differential relation 

Mdy _ dx _-, 

~ 



connecting x = $n\u, K) and y = su 2 (u/M,\), 



THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 317 

we notice, by 139, that the absolute invariant J is unchanged ; 
so that, according to 68, there are six values of I, given by 



= k, 



1 



l-fc, 1-4; 



fc-1 &> !-& A; 

and six corresponding linear transformations, in which 
Afi aK+bK i 



.(53) 



cK+dK i 



and be ad=l; (54) 



a, b 
c,d 



1 
1 




1 1 

1 




1 

1 1 



1 
1 1 



1011 
10 1 



mod. 2. 



295. But if we change to Jacobi s form by the quadric 
transformation, which changes x into x 2 , and y into y 2 , then 
Mdy _ dx _ , , 

" 



and now, forming according to 75 the invariants g. 2 , (/ 3 , A, and 
/ of the quartic 1 x 2 . 1 /c 2 # 2 , 

& = 



12 



216 



16 



and 



.(56) 



lOS^l-Jfc)* 

Professor Klein writes >; 4 for k or /c 2 , and calls r] the Octa 
hedron Irrationality ; and now the absolute invariant being 
unaltered by a linear transformation, 



)41 



108Z(1 -O 4 

and the roots of this equation in I are found to be 



(57) 



. 

giving the six corresponding linear transformations of Abel 
((Euvres, I, pp. 459, 568). 

In the reductions of Chapter II. that linear transformation 
has been chosen which makes k or I positive and less than 
unity, and also gives a real value to the multiplier M. 

The corresponding values of the multiplier are given by 



the linear transformations being, as may be verified. 
y=x, rfx, :r-_ 



+ rjX* 



I + irj 1 + lt]X 



318 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 

Landeris Transformation of the Second Order. 

296. The point L ( 28) in figs. 2 and 3 has been called 
Landens point, because of the use made of it by Landen 
(Phil. Trans., 1771, 1775) for his transformation, important 
historically as the first case investigated of the Transforma 
tion of Elliptic Functions, being the Quadric Transformation, 
or of the second degree. 

The ratio AD/AE being sin 2 | a or /c 2 , while EL/EA = cosa 
or K ; therefore, if C is the middle point of AD, 
LG AL-AC AE-EL-^AD 



GA AG 

_ 1 cos a \ sin 2 a _ (1 cos Ja) 2 _ 1 cos 

J sin 2 a 
The ratio LC/CA is denoted by X ; so that 



21 
~ 



a. 



(l-OM */* =(l-W, and / cX / = 
different forms of the modular equation of the second order. 

Still denoting the angle ADQ in fig. 2 by 0, we denote the 
angle ALQ by i/r; and now ( 28) since the velocity of Q 
is 7i(l+/c x )^Q, perpendicular to CQ, therefore the component 
velocity of Q, perpendicular to LQ } 

LQ d^/dt = n( 1 + K )LQ cos LQG, 
or d\ls/dt = n(l+ K )cos LQG. 



sin LQG LG , , 
But since : = 777, = X, therefore 
sin I//- CQ 



sin LQG= X sin ^, cos ZQ<7= VC 1 - x sin V) = A(^, X) ; 
and d\lsldt = n(l+ K )&(\Ir, X), 

or ^=am{(l+ic M X} ......................... (60) 

Now, since the angle LQC=2(j> \fs, therefore 

sin(20 -<//) = A sin i/r; ................................... (61) 



/ __ = 

~ nr tan 



or tanx^ = \ /- f t .............................. (03) 

1 K tan 2 < 



sin i/r = (1 H-^sin cos 
as in equation (92), 67. 

Putting nt = u, (!+K) nt = v, then sin^> = sn 
and we obtain the formulas (90) to (98) of 67. 



THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 319 

297. Landen starts with the relation (61) ; so that, differen 
tiating logarithmically, 

cot(20 \fs)(2d<p d\{/) = cot \lr d\[s, 
2 cot(20 ^)c?0 = {cot(20 \I 

sin 20 d\fs 
sin \jr sin(20 



sin 20 cosec \fs ~ cos(20 \/s) 

Now cos(20 - ^) = ^/(l - X 2 sin 2 x/r) = A(^, X) ; 

while sin 20 cot i/r cos 20 = X, 

cot \js = cot 20 -f X cosec 20, 
cosec V = 1 + (cot 20 + X cosec 20) 2 , 
sin 2 20 cosec 2 ^ = sin 2 20-f(cos 20 + X) 2 
= 1 + 2X cos 20 + X 2 
= (l+X) 2 -4Xsin 2 0, 

or sin 20 cosec \lr = (l+\)J(I- K 2 sin 2 0) = (1 + X)A(0, AC), 
where /c = 2 x /X/(l+X) ; so that, finally, 

c?0 _K1+W d+ _(l+QcZ0. 

~ 



so that, if = am(n, K), then ^/r = am{(l+/c / )ni, X}, and the 
angle \/r may be made to represent pendulum motion on the 
circle CRL, on CL as diameter, LQ meeting this circle in It. 

The velocity of R will then be due to the level of L , a point 
on CE produced, such that CL = CL/X 2 ; and now we find that 

EL = CL -CE=EL, 

after reduction, so that L and L are the limiting points of the 
circle AQD with respect to the horizontal line through E\ but 
now the value of g in the motion of R on the circle CRL must, 
in accordance with 20, be reduced to J<?(1 O 4 - 
. L Q_L D_EL+ED_ K + K - 2 _I+ K 

LQ ~LD -EL-ED- K - K *-T^" 
so that ( 28) the velocity of Q is 

n(l+ K )LQ, or n(l- K )L Q ................. (65) 

The period of R in the circle CRL is half the period of Q in 
the circle AQD; so that, if A denotes the real quarter period 
of the elliptic functions of modulus X, 

^ or (l+A)A = Jf. . ..(66) 



320 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 



298. Conversely, as in 123, we can express the elliptic 
functions of modulus K and argument (1 + \)v in terms of the 
elliptic functions of modulus X and argument v ; or starting 
with the motion of R, we can deduce the motion of Q. 

But considering the motion of Q as defining in a similar way 
the motion on a larger circle, to a larger modulus y, we change 
X into AC and K into y, where 
1 y , 1 K 



and now, from 123, 



,. . x 1 AC sn 2 (u, AC) 

dn(l+/c.u, y ) = - -^-M> 



/., v 

cn(l + K . w, y) = 



1 + AC sn 2 (u, AC) 

cn(u, Ac)dn(u, AC) 

1 + K sn 2 (i6, AC) 



called Landens Second Transformation. 

With x = sn(u, K ), y = sn(l+K .u,y\ v/here y 

then 



.(68) 



AC), 



and 



- y 2 . 1 - yV ) 

Or, with x = dn(u, AC), 2/ = dn(l + AC . u, AC), 



y = 9* 

= 2/c -s-F, 

1-2/ =2(1 -a; 2 ) s-F, 



leading to the differential relation, (3) of 35, 
dy _ (1 + K)dx 



(69) 



(70) 



THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 321 

299. Denoting by T the real quarter-period of the elliptic 
functions to modulus y, then x = 1 makes y = 1, or u = K makes 
(l+ K ).u = T ; so that 



or (66) (l + X)A = /i=4(l + y )r ......................... (71) 

Also, A , K , r" denoting the corresponding quarter periods to 
modulus X , K, y, the imaginary transformations of 238 show 
that, with iu = v, 



, N/ v cn(v, K)dn(v, K ) 
cn(l+,c .v, X)= -.- 



cn(l+/c .v, y ) = 



3 /i , / >/N 1 /c sn 2 (v, ic ) 

dn(l + K . v, X ) = . , , z , (> 

l+rsn*(v, /c) 



1 (1 /c)sn 2 (t , /c ) 
- 



so that A = (1+OK , r 

or i(l4.X)A = K / = (l + y )r ; .................. (73) 



and therefore 1 A! = ^ = 2^. ... (74) 

J A -LV 1 

An inspection of Landen s formulas shows that the dn func 
tion has always a rational Quadric Transformation. 

Mr. R. Russell shows (Proc. L. M. S., XVIII.) that the 
general rational quadric transformations which reduce 

dx/JX to the form 
are always of the form 



P v P 2 , P 3 denoting the quadratic factors of G, the sextic 
covariant of X ( 160). 

Thus if X = 1 - x 2 . 1 - K*X*, 

the sextic covariant may be written 



leading to Landen s transformations, given above. 

G.E.F X 



322 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 

300. Landen s Transformation is useful, as employed by 
Gauss, for the numerical calculation of K ; for if we put (fig. 2) 

LA = a, LD = b-, and GA = a v GL = J(a* - b*) = |( - &) ; 
then o 1 = J(a + 6), b, = J(ab); and K = b/a, X^bja^ ...(76) 
Now, denoting \js by <f> v and X by K V equation (64) becomes 
2^0 dfa . (?7) 

^(tt 2 cos 2 + 6 2 sin 2 0) ^/(a 2 

while 0! = TT, when 

so that 



i + 6 1 2 sin 2 ] 



_ /*" _ % 

y JW**fa+bfafy 



or = l aa fl = K 1 (l+ Kl ) ...................... (78) 

Continuing this process with 15 a lf and b lt so as to obtain a 
continuous series, given by ( 296, equation 62). 

= - tan <p n , 



, 6n+l = V( a n&n); ............. (79) 

then a rt and 6 W tend to equality ; so that, putting 
a= & =/* and =^ 



/!L_ 

J ^/(a 2 cos 2 





r*r 



or 

a 



._, .-., . -*) (8<>) 

r=l r=l 

Denoting the modular angle of K H by n , then 
Kn+l = sin (9 n +i = tan 2 J$ n ; 

COS 9 n+ i = SeC 2 J0 n -v/( COS ^n) 

and 1 +/CH+I = sec 2 J0 n = // n ^x > 

so that 

jfir= jTrsec OVC 008 cos ^ cos 2 cos 9 3 ...), (84) 

a formula suitable for the logarithmic calculation of K. 



THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 323 

The Transformation of the Third Order, and of higher 
Orders. 

301. According to Jacobi s method, the transformation may 
be written 



1 _i_<?/ i i ->-v i i ~/ v"*y 



connecting x = sn.(u t K) and y = sn(u:M, X) ; and then 

i q + 1 + aV _ x I-x*/a z 
l + (a 2 + 2a)B 2 ~ M l-K*aW~ 
so that l/Jf=2a+l, 

- 



leading to the differential relation 

* 



We shall find that, expressed in terms of a, 



= 



and ,,_(l-a)(l+a) s ,, _ ( 1 + aX_l - a) 3 

"" 



so that 



leading to the Modular Equation of the Third Order. 



We shall also find that this transformation ma} be written 
1 cn(u/M, X) _ 1 en u /a -f 1 + a en u\ 2 
1 -f- cn(u/M, X) 1 + en uAa -f 1 a en uJ * 

l+dn(u/4fl A)~~l+dn iAa+1-dn J ^ 88) 

As a numerical exercise the student may work out the case 



of a 

In Legendre s notation, with ic = siu0, ?/ = sinx/r, he finds 
tliat these relations are equivalent to 

The Transformation of the Third Order was the highest to 
which Legendre attained, until it was pointed out by Jacobi 
in the Astronomische Nachrichten, No. 123, 1827, that Trans 
formations exist of the fourth, fifth, or any other higher order, 
as already explained. 



324 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 

tt 

, 



Thus the transformation of the fifth order may be written 
1 _ x n _ 



and of the seventh order 



and so on. 

302. When the transformation of the third order in 157 is 
employed for the reduction of the integral in equation (6), 227, 

then s 3 =-E" 3 /P 2 , (92) 

where P = p 3 - 3p 2 sin 2 a + 3p, (93) 

and ./T => 2 cos 2 a+psin a 1, (94) 

as in equation (27), 233 ; so that ^=0 and 8 = at the points 
of minimum velocity. 

Now, with this substitution of 157, 

s = p(yx/w 2 ; 0, A), (95) 

where A = 4 - 3 sin 2 a = 27# 3 , (96) 

( 228) ; and denoting 

/ao 

-V<1*> 

then 0Q 9 = 0, p JQ 2 =-/J f A > and # 9 Q 9 = i-Tr^S ( 293). 



Again (157), v ( 
where J=p*(3 sin a - 2 sin 3 ) - 3^ 2 (2 - sin 2 ) + 3p sin a - 2, 

and J+P /v /A = 2{i(sina + /v /^)P- 1 } 3 

/-P /v /A = 2{Ksina-v / A)p-l} 3 ............. (97) 

Now from 233, 

^/A = cos a(tan /3 + cot {$), 

|(sin a 4- >v/A) = J cos a(tari a + tan /3 + cot ) 8) = cos a tan /3, 
J(sin a x/ A) = J cos a(tan a tan /3 cot /3) = cos a cot $ 

. ., sin 6 

while p = -. - -- 

cos(a 6) 

Therefore 

(^; u> -A) -^112.2 



5 0, - 

cos a tan ^ sin cos(a 0) _ tan(/3 0) _ tan (j> 

~ cos a cot /3 sin cos(a 0) ~ tan /3 ~"tanj 

_V (u- 0, .g 3 )-^ 



( 234) a curious result of this transformation. 



THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 325 
Again, since $ %a). 2 = tf ^wy we may put 



and then, making use of relation (17) of 229, 



<r(ia> 2 - u)o- 2 ( ^. 2 + w)P(il2 + u) 

by means of (K) 200, and the relation #?f o.\ 2 = ; and this 
again, by equation (CC) 279 and by 293, reduces to 



_ fl r(}Q i - tt ; 0, -A) .. 
- ; 0, -A) 6 



Transfoi^mation of the Theta Functions. 
303. Taking the function, as defined in 263, 265 in the 
factorial form, 

0(x,q) = <j>(q)Il(l-2q 2r - l cos2x + q* r - 2 \ ...... (100) 

r = l 

where <f>(q) is a certain function of q which 264 shows can be 
written <f>(q) = H(l - q- r \ .................................. (101) 

then changing x into nx, and q into q n , 
0(nx, q n ) = (p(q n )Ii(l-2q-" -- l c 



II_ S f[ {l-2(/ 2r - 1 c 

? = ! = o 

(by Cotes s Theorem of the Circle of 270) 

f i.( f1 n\ s = n-l 

= {Sl?> a!+8 /n ?) ...................... (102) 

Similarly, with yu = l, 2, 3, 



Forming the quotients, and writing x for %7rU/K, then ( 263) 
1 0-.X 



and thence we obtain the formulas for the Transformation of 
the Elliptic Functions of 283. 



326 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 

Similar considerations will show that, when q is changed to 



where yu = 0, 1, 2, 3 ; this is left as an exercise (Enneper, 
Elliptische Functionen, 38). 

EXAMPLES. 

1. Prove that a transformation of the fourth order is 
1 1 x lKX/Ix 



+y 
and prove that the relation between X and K is then 



and M= 

2. Prove that, by means of the substitutions 

cosh ^u sinh <& 
=-r-r , 

u + sinh u cosh 

cosh ^u sinh < 



or sin = . 



. , t rn ~i 
sinh JM + cosh iucos 



u cosh 
dO 



= sech 





cosh 



(cosh u + sinh u cosh 



o 



1 .3. 5...2m-l 1 /* ^ (sinh ^) 



-3.. . 2u - 2m 

o 



3. Prove that, with the homogeneous variables # 1} # 2 of 155, 
and writing X l for dX/ dx^ X 2 for dXj dx^ the general cubic 
transformation which reduces dx/ +JX to the form 



is of the form z = (lX 1 + mX 2 )/(l X l + m X 2 ) (ex. 8, p. 174). 

Prove also that the general quartic transformation may be 
written z = (lX + mH)/(l X + m H), 

where H denotes the Hessian of the quartic X ( 75). 

(R. Russell, Proc. L M. 8., vol. XVIII.) 



THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 327 
4. Prove that (Cayley) 



satisfies the relation 

dy pdoc 



Modular Equations. 

304. In the Transformations of the n th order, which con 
nect the Elliptic Functions of modulus X with those oi 
modulus K, and make r = q n , or q lln , or a)Pq l/n ( 285), 
A i K i 1 K i 2pK+K i ,, .aK+bK i nn ^ 

" or - or ~ or s enerally (106) 



where 6c ad = n, 

the Modular Equation, which determines X in terms oi /c, is of 
the (?i+l)th order, as already stated, when n is prime, and 
has two real and n 1 imaginary roots. 

We shall content ourselves with merely stating the Modular 
Equations of simple order, connecting K, X and AC , X , adopting 
the form and classification employed by Mr. R. Russell in the 
Proc. London Math. Society, Vol. XXT. 

CLASS I. 71 = 15, mod. 16 ; 



Q = 4 

E = 4 

7i. = 15, P 3 - 



CLASS II. w = 7, mod. 16; 



= 7, P = 0, or 4/(/cX)+ / 4/(/c X / )-l, (Guetzlaff). 

-23, P-3 = o, or 4/( / cX)+4/(/c / X / ) + (256 / cX/X / ) TV 
= 71, P 3 -4E4(P 2 -Q) + 2PPl-P-0. 
= 119, P 8 -J^(7P 5 -28P 8 Q-hl6PQ 2 )+ J R 8 (...)...=0. 



328 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 
CLASS III. n = 3, mod. 8 ; 

p= 

Q = J 



n = 3, P = 0, or J(K\) + J( K \ ) = I, (Legendre). 

7i=ll, P-/# = 0, or 



7i = 43, P u +...=0. 
7i = .59, P 5 + ... = 0. 
^ = 83, P 7 +...=0. 

CLASS IV. ?i = 1, mod. 4? ; 



w = l, P = 0. 

7i = 9, P 6 - 1 4P 3 J2 + G4PQ^ - 3-R 2 = 0. 

71=17, P 3 -H lop2 -64Q) + 26JAP + 12/2 = 0. 



7i = 29, P^(P 2 + I7R*P- 9R%) 

JR^gP 2 - 64Q - 
7i = 37, 
71 = 53, P2{P 4 + ^(413P 3 -2 1(5 PQ)+. 

305. According to Professor Klein (Proc. L. M. 8., X. ; Math. 
Ann., XIV.) these Modular Equations are replaced by relations 
between the absolute invariant J and its transformed value / , 
by the intermediate of quantities T and T , such that J is a 
certain function of T, and J the same function of T , and now, 
7i = 2; /:/-!:!= (4 T -1) 3 : (T- l)(8r + l) 2 : 27r, 

TT =1 (60). 
7^ = 3; /:,/-! :1= ( T - l)(9r-l) 3 : (27r 2 - 18 T - I) 2 : -64 T , 

TT=1. 

71 = 4; J:,/-l:l= ( T 2 + 14r+l) 3 : 

( T 3-33 T 2 -33 T H-1) 2 : 108r(l-r) 4 , 



THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 329 

w=5; J:J-l:l= (-r- 
: ( T -2_ 

TT=125. 

71 = 7; J:J-1:1 = (T 2 +13 T + 49XT 2 

7()T-7) 2 : 1728 T| 



72 = 13; J:J-l:l= ( 

: (T 2 +6T+13)(T 6 +10 T 5 +46T 4 +108T 3 +122T 2 +38T-1) 2 : 1728r. 
rr = 13. 

The Multiplication of Elliptic Functions. 

306. If we perform the second real transformation upon the 
first real transformation, we obtain a transformation of the 
order n 2 , leading back again to the original modulus K ; because 
the first real transformation changes q into q n , and the second 
real transformation changes q n back again to q. 

We then obtain the elliptic functions of argument 

u/MM t =nu, since M = K/n-A, M r =A/K f 
in terms of the elliptic functions of argument u, by a trans 
formation of the order n 2 , and thus obtain the formulas for 
Multiplication of the argument. 

Thus multiplication by 2 or 3 can be obtained by two suc 
cessive transformations of the second or third order ; and so on. 

Knowing that the order of the transformation is n 2 , we 
infer in Abel s manner the factors of the numerator and 
denominator of the transformation, involving the modular 
functions, the elliptic functions of the 7ith part of the periods. 

Thus we infer, with the notation of 258, that, for an odd 
value of 71, 

snrm = UjV, ........................................ (107) 

where U = n sn u II IlYl - ~ 

v= 



where m, m =0, 1, 2, 3,..., $(n-l); 

the simultaneous zero values of m and m being excluded. 

as denoted by the accents, so that the number of factors is 



330 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 
Combining the factors by formula (7) of 137, 



sn(^-O/^), (108) 
where A is a constant factor ; and this may be written 

sn?w = 4nil8n(% + Q/w); ...................... (109) 

where m, m = 0, 1, 2, ..., +(w-l); 

the simultaneous zero values of m and m being now admissible. 
Similar considerations will show that 

cn?iu = nncn^-f Q/TI), ........................ (110) 

dnwtt=0nndn(tH-Q/tO ........................ (Ill) 

To determine the constant factors, change u into u + K or 
u+K i, when we shall find (Cayley, Elliptic Functions, 368) 
4 = (-1)^-1)^2-^ B = ( K /K ^- I \ C= (l/ic )K n *- l \ 

By taking in 259 a rectangle OA n B n C n , in which (M 7l = ?ia > 
OB n = nb, and therefore containing ?i 2 elementary rectangles, 
we obtain a physical representation of the formulas (109), 
(110), (111) for Multiplication of the argument by n. 

Writing u/n for u, and making n indefinitely great, we 
deduce in a rigorous manner the doubly factorial expressions 
for sn u, cnu, dnu in (1), (2), (3) of 258. 

Again, by putting /c = or /c = l, the student may deduce as 
an exercise the trigonometrical formulas for the resolution of 
the circular and hyperbolic functions into factors. 

(Hobson, Trigonometry, Chap. XVII.) 

The Complex Multiplication of Elliptic Functions. 

307. When K \K ^/D, and D is an integer, we may sup 
pose the multiplier n resolved, by the solution of the Pellian 
equation, into two complementary imaginary factors, so that 



and now the multiplication by n can be effected by two suc 
cessive multiplications by the complex multipliers a-\-ib^/D 
and a ib^/D, each leading to an imaginary transformation of 
the Tith order, not changing q or the modulus K. 

(Abel, (Euvres, I, p. 377 ; Jacobi, Wcrke, I., p. 489.) 

The first requirement then in Complex Multiplication is a 
knowledge of the value of K for which K jK= *JD ; and this 
is found by putting K = X , K = X in the corresponding Modular 
Equation of the order D ( 304). 

The equation is now, according to Abel, always solvable 
algebraically by radicals ; so that, returning to the question of 



THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 331 

the pendulum in 1 5, it is possible to determine by a geometri 
cal construction the position of two horizontal BB , W, as in 
fig. 1, cutting off arcs below them, such that the period of swing 
from B to B is ^JD times the period from b to V. 

Thus the Modular Equation of the second order being 
written X = (1 - K )/(! +* ) 

we find, on putting /c = X, 

X 2 + 2\ = l, or \ = J 2-1, when A /A = ^/2. 

Putting K = \ , K=\ in the Modular Equation of the third 
order ( 304), 

2 /v /( /c/c ) = l, or 2/c/c = i=sini7r, when K /K=J3; 
so that the modular angle is T W or 15. 

When K /K=2, /c = U/2-l) 2 (71); 
obtained by putting r/r = l, y = y =J^/2 in 298,299. 

When K /K = j5 t 2 KK = Jo-2~ 4/(2^ ) = KV 5 - 1 X 
or (2/c/c )~^-(2/c/c f = 1. 

When K IK=J*l t 2^/(^ ) = l, 2 = , 4/(2 )=i- 

Collections of these singular moduli required in Complex 
Multiplication are given by Kronecker in the Berlin Sitz.> 
1857, 1862, in the Proc. L. M. S., XIX., p. 301 ; also by Kiepert 
in the Math. Ann., XXVI., XXXIX., and by H. Weber in his 
Elliptische Functioned, 1891. 

308. In the expression of y = sn(a-\-ib All /D}u as a rational 
function of x = snu, leading to the differential relation 
Mdy dx , 



Jacobi finds (TferA;e, t. L; de multiplicatione functionum 
ellipticarum per quantitatem imaginariam pro certo quodam 
modulorum systemate) that we must restrict a to be an odd 
integer, and b to be an even integer ; but these restrictions 
disappear if we work with the en functions ; and we can 
even suppose that 2 and 26 are odd integers. 
Let us determine then the relations connecting 

x = cnu and ?/ = cn J( 
so that I/M= -J + J i 

leading to the differential relation 
dy ( 



where C = K/K, the cotangent of the modular angle. 



332 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 

If D = 4<n 1, and we denote (K+K i)/n by w , we shall 
then find that, when n is odd, 



c 



ic ic 

but, when ti is even, 



. 
-- (1 



x 

T 



c c 



ic 

The arithmetical verification for the simple cases of D = 3, 
7, or 15 is left as an exercise for the student (Proc. Cam. 
Phil. Society, Vol. V.). 

Formulas (112) and (113) are inferred by putting 

(1) y-1, 

when -J( - 1 + i^/D)u = 2mK + Zm K i (m + m even) ; 
and then u = fan K (m + m^co, x = en 2? ft). 

(2) ?/=-!, 

i -( - 1 + iJD)u = 2mK+ Zm K i (m + m odd) ; 
and then x = cn(2r l)w. 

(3) y = ic, 

i (w + m odd); 



(4) y=-ic, 

\(-l+iJD}n = (2m+l)K+(2m +l)K i (m+m x even): 
and then x= cn(2r l)co. 

309. When D = 4>n + I or 1, mod. 4, the relation connecting 
oj = en u and y = en \ ( 1 + i^D)w cannot be rational ; but Mr. 
G. H. Stuart has shown (Q. J. M., Vol. XX.) that it may be 
written in the irrational form 



THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 333 

where o> - (K+K i)/(2n + 1), 

a transformation of the order n + i ; and this is equivalent to 



2 

-F, 



F= (i_?)n(l+- -I ; ...... (114) 

V ic/ I cn(2?- l)o>J 

this is inferred in the same manner as formulas (111) and (112). 
For instance, with n = 0, D = 1, and K = J^/2, c = 1 ; 

i/ -I , -\ //-\ //^ + cn u\ 

en J( 1 +i)u = *J(i)J( ^ J - 

y \ \i en u/ 

equivalent to, with u = (1 -f i)v, 

/1 .. .1 icn z v 

cn(l \)v = ^^-:. ,- 

1 + i cn 2 t 

With w = l, D = 5, 2/c/c = x / 5 - 2 ^ = 



/c _v/5 + l, U5 + 1. 

V c 2 V 2 

and en i - 1 + iou 




where a = en i(lf+ K i). 

310. Generally in the expression of y = $ulM as a function 
of # = #?&, where 

w /w or K i/K=J(-D), 
and the multiplier 1/Jlf is complex, of the form 



it is convenient to consider four classes of 1). 

Class A, D = 3, mod. 8 ; 

Class B, D = 7, mod. 8; 

Class C, D = l,mod. 4; 

Class D, D = 2, mod. 4; 

the class for D = 0, mod. 4, not requiring separate consideration. 
It is convenient also to consider the discriminant D ( 53) as 
negative ; a change to a positive discriminant being effected by 
the method of 59 ; now w Jw. 2 = i^/D. 

We can also normalize the integrals ( 196, 252) by taking 
g* - 27# 3 2 = - 1, so that g, = J/( - J). 



334 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 
CLASS A. D = 3, mod. 8 = 8p + 3 or 4>n-l, if n = 2 



The relation connecting x and y can be written in one of 
the three equivalent forms 



-eJ IL {x-^ 1 + 2ra) 3 M)} 2 - F, 
V= H{x-p(2r 

leading to the differential relation 

Mdy _ dx 



This verifies in the particular case of p = 0, when 



and then 

This is the simplest case of Complex Multiplication, 
meationed in 196, and employed in 227 in the determina 
tion of the Trajectory for the cubic law of resistance. 

The form of the general transformation is inferred from the 
consideration of the series of values of u which make 

y or <p(ujM) e v e z , e%, and GO . 
(i.) When y = e lt 
u/M= 



2n 



so that ic or $>u = e x or 
(ii.) When y = e 2 , 



= e, or 



THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 335 

(iii.) When y = e s , 



u= g 

<pu = e v or ^w 2 4-r ft) 3 ?i or 
(iv.) When y=ac , 



it = 2gw 2 + 2 / 
and $m = j0(2ro> 3 /?i). 

Hence the form of the Transformation is inferred. 
By addition, we find 

n>n A , 7> n - 1 i A fii - 2 

j re) ^O ~~ X1 1 iXy ~f XI ,-)>Aj . . 

^ i ji I ^ _ _ ^ __ 

(o;P-(7 1 ^- 1 + (?^- 2 ...) 2 
where ?i = 2p + l; and we shall find that xl 1 = 2(r 1 ; and the 



xl s and (? s are symmetrical functions of e v e. 2> e%, and there 
fore functions of g< # 3 or J\ while ^ has the same significa 
tion as in 293. 

By employing the Modular Equations given above, or 
employing Hermite s results (Theorie des equations modu- 
laires), we find 
=3, J=0, 9t = o, 



n T -* 8 

=11, J=-, g,= g= 



) A _^ , . 

/i -"-2 -jo .3~ 

= 3, 9s 



these values of J. , A 3 , A, A 5 were calculated by Rev. J. 

Chevallier, Fellow of New College, Oxford, who has also 

verified the case of D = 1 1 . 

D =27, J=-2 9 x5 3 --3 2 , etc. 

D =35, & = f /v /5{iU/ 

D =43, /=-2x5, 

^ = 3x7x^43 (Hermite). 



73^43, etc. 
=51, /=-64.(5 + Vl7) 3 U/ l7 + 4 ) 2 (Kiepert). 



336 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 



D =67, ,/=-2 9 x5 3 xll 3 , # 2 = 

#, = 7x31x^/67 (Hermite). 
D =163, /=-2 12 x5 3 x23 3 x29 3 , J(g 2 + l) = % x 7 x 11, 

(/ 3 = 7xllxl9xl27x /163 (Hermite). 

CLASS B. D = 7 t mod. 8 = 8^ + 7 = 4^-1, if n = 2p + 2. 
The relations connecting y==p(u/M) and o; = pu, where 



are found, in a manner similar to that employed in Class A ; 
- ej(x - e 9 ) Yl \x - p( 2 + 2n*> : } 2 -f- F, 



2 4- 7, 

r-O 

Jf 2 nf {^ - pfo + 2ro,} 2 -j- F, 



r 

As simple numerical applications, 



/> = 1 5, ^//c/c = sin 18 ( Joubert). 

In these cases the Jacobian notation is almost more simple, 
as given in 308. 

CLASS C. D = 1, mod. 4 = 4n + 1. 

The relations connecting x = <pu and y = tfu/M), where 



cannot now be rational ; but, according to Mr. G. H. Stuart, 
we can express the relations in the irrational form 



a relation which may be said to be of the order n + \ and 
this is equivalent to 







THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 337 

CLASS D. D an even number. 

In this class the simplest function to employ is the sn func 
tion ; for instance, with 

K \K= ^/2, then K = v /2 - 1 ; 

sn-u 

~~" 2 i) " 

and 



where <o = J( K iK ) ; 

leading to the equations 




1 K.y _ 1 #/! + K x sn wV 2 
1 -f Ky 1 + 5\1 /ca? sn o>/ 
connecting ic = snu and y = su(l-t- 

/ 2 i 
Also = 2 = ^ 



These transformations show that it is not possible to express 
cn(l-M x /2)i6 in terms of cn-u-, or dn(l + ? v /2)u in terms of u, 
by a rational transformation. 

With K /K=2, then /c = ( v /2-l) 2 (71), 
and the relation connecting x = snu and y = sn(l + 2i)u may 
be written 

/ /y.2 \ / ^2 \ 

sn*4w/ 



(l-A 2 3n 2 2a))(l-/c- --> 
w here o> = i( A" iK ) ; 

equivalent to the relations 

/ T \ 7 / V 

n - ^_\ /ij a 

l-y 1-lrtBJ 



1- 




sn z 

so that cn(l + 2i)i6 has a factor dnw, and dn(l + 2/)w, has a 
factor en u. 

G.E.F. Y 



338 THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 



When K /K=J6, then K = (*J3- J 2)(2- 
and the corresponding relation between snu and 
to be written down is left as an exercise. 

(Proc. Gam. Phil. Soc., Vols. IV., V.) 

It can also be shown, in the preceding manner, that the 
relation connecting x = <pu and y = $(ujM) where 



i\}\d D is an even number 2m, can be expressed by the relations 



-c, A = M \x-,g H -pi--<>2 / + v > 



As numerical exercises, we may take 
(i.) D = 2, when ^ = 30, $r 8 = 28, ^=- 
(ii.) D = 4, when </ 2 = ll, $r 8 = 7, G^- 

oil. In conclusion we may quote from Schwarz some 
general remarks on doubly periodic functions. 

Every analytic function </>u of a single variable u for which 
an algebraical relation connects <f)(u + v) with (fru and </>?; is 
said to have an Algebraical Addition Theorem ; and then <f>u 
must be an algebraical function of 0u (Chap. V.). 

Every such function is then an algebraical function, or an 
exponential function (circular or hyperbolic function), or an 
elliptic function, which can be expressed rationally by pu and 
V u (Chap. VII.). 

Elliptic functions are doubly periodic. A function of a 
single variable cannot have more than two distinct periods, 
one real and one imaginary, or both complex. For if a third 
period was possible, the three sets of period parallelograms 
obtained by taking the periods in pairs would reach every 
point of the plane, so that the function would have the same 
value at all points of the plane, and would therefore reduce to 
a constant (Bertrand, Calcid int&jral, p. 602). 



THK TRANSFORMATION OK KLLIPTIC FUNCTIONS. 

Abel, in generalising these theorems, was led to the discovery 
of the hyperelliptio and Abelian functions. 

Thus if X in 169 is of the fifth or sixth degree, we obtain 
functions of 2 variables and 4- periods ; if of the 7th or 8th 
degree, of 3 variables and 6 periods; and generally, if X 
is of the degree 2p + l or 2p + 2, there are p variables and 
2p periods ; but this would lead us beyond the scope of the 
present treatise, and the reader who wishes to follow up this 
development is recommended to study Professor Klein s articles 
u Hyperdllptische Sigmafunctionen" Math. Ann., XXVII., 
XXXIH., etc. 



APPENDIX. 

I. The Apsidal Angle in the small oscillations of a Top. 

The expression given by Bravais in Note VII. of Lagrange s 
Mdcanique analytique, t, II, p. 352, for the apsidal angle in 
the small oscillations of a Spherical Pendulum about its lowest 
position is readily extended to the more general case of the 
Top or Gyrostat, if we employ the expression on p. 201, 242, 
as the basis of our approximation. 

We divide the apsidal angle ^ into two parts, ^ and "^ 
such that i ^ l = ar] l a) l ^(ij 

^ f2 =bt h -w^b- 

and now put a = o) 3 sa> 3 , b = w l + qcD s , 

where q and s are small numbers; so that, expanding by 
Taylor s Theorem as far as the first powers of q and 8, we may 
put ^a 



and now, by means of Legendre s relation of p. 209, 



But, from equation (B), 51, 

vu e;, 
- ^ = 2 = 1 



so that, integrating between the limits and oo v 



o 
or ^ + e l ta l = V( e i ~ s)^ (Sch warz, 29). 

Also (51) (^-^3)0)!- ^/(^-e.^K- 
so that i^! + <? 3 wi = V(^i "" e z>(K ~~ ty 

and therefore i^ = \i-jr + ^ 3 ^/( fl i 



340 



APPENDIX. 341 

But, from 210, when a and ft are very nearly TT, their 
approximate values are given by 



since f // a = 2(e? 1 e 3 )(e 2 e s)> 

and K 2 = ^A ^ = Vzi2( 52 ); 



and therefore 
Also ( 210) 



G-Cr 



; 
/C 2 



so that (^ - e 3 )q V - - T -2 cot2 i cot 2 



2 



_ 

Therefore ^ ^ JTT + 3 jc cot Ja cot 



E 



But, ultimately, when /c = and /c r =l, 
then ^=i7r, and lt(/i _^)/ AC 2 = i 7r (11,170); 

so that Mfj ^ JTT + ITT cot Ja cot i/3, 

7r cot - a cot 3. 



This reduces for the Spherical Pendulum, in which (7? = 0, to 

^ S JTT( 1 + f cot Ja cot J/3) S $ir(l + 1 sin a sin /3), 
when a and ft are nearly TT, thus agreeing with Bravais s result. 

When a = 7r and 6r-f-(7r = 0, this approximation fails; but 
the student may now prove that the apsidal angle is 



This will be the apsidal angle when the Top is spinning in 
the vertical position with small angular velocity / , and is then 
struck with a slight horizontal blow. 



342 THE APPLICATIONS OF ELLIPTIC FUNCTIONS. 

II. The Motion of a Solid of Revolution in infinite friction- 
less liquid. 

The reductions of the Elliptic Integral of the Third Kind 
in 282 in consequence of the relation 

a + b = u> a , 

in connexion with the Top and Spherical Pendulum, are useful 
also in constructing degenerate cases of the motion of a Solid 
of Revolution in infinite liquid, as mentioned in 211. 

We refer to Basset s Hydrodynamics, Vol. I., Chapters 
VIII., IX., and Appendix III., also to Halphen s Fonctions 
elliptiques, II., Chap. IV., for an explanation of the notation ; 
and now T the kinetic energy of the system due to the 
component velocities u, v, w of the centre of the body along 
rectangular axes OA, OB, 00, fixed in the body, 00 being the 
axis of figure, and to component angular velocities p, q,r about 
OA, OB, 00 is given by 

T=$P(u* + v*)+$Rw z + } 2 A(p* + q*) + $Cr* (A) 

(to which the terms 

P (up + vq)+P wr 

may be added in the case of a body like a four-bladed screw 
propeller, or like a rifled projectile provided with studs or 
spiral convolutions on the exterior). 

Then the Hamiltonian equations of motion are 



d 


VT 

- T 


VT 


VT 


v 




m 


dt 
d 


Vu 

VT 


VT 


VT 
_i_ Y __ 


-y, - 




...(2) 


dt 
d 


Vv ^ 

Q 


VT 


Vu 
VT 


-z, .. 




...(3) 


dt 
d 


Viv " 

VT 


Vu 


VT 


VT 


v L 




dt 
d 


Vp 


VT 


VT 


VT 


VT 

W -M 




dt 
d 


Vq P 

VT 
a 


VT 


T Vp 

VT 

4-99 


VT 


Vu 
+**=*. 


. (6) 


dt 


Vr q 


Vp 


+r> -d<j 


Vu 


Vv 





When no forces act, so that X, Y, Z, L, M, N vanish, then 
equation (6) shows that Or or r is constant. 

Multiplying equations (1) to (6) by u, v, w, p, q, r in order, 
adding and integrating, shows that T in (A) is constant. 



APPENDIX. :U:{ 

Multiplying (1), (2), (3) by ||, g, g adding and in- 
tegrating, proves that 

/32V , /37V /32V . 

I - - ) + ( ) + ( - ) is constant ; or 

\du/ \dv J \dwJ 

pi(u* + v~) + RW = F 2 , ..................... (B) 

F being a constant, representing the resultant linear momentum 
of the system. 

Similarly, it is shown that 

-dT^T -dT^dT 3T3T. 

- is constant : or 



r=G, ................... (C) 

where G is a constant, representing the resultant angular 
momentum of the system. 
From equations (A) and (B), 

A( p* + q-) = 2T- 6V 2 - Ru A - P(u 2 + v 2 ) 



and, from equation (3), 



so that ^y or ^^(; is an elliptic function of t. 

Taking the axis Oz in the direction of the resultant impulse 
F, and denoting by y p y 2 , y 3 the cosines of the angles between 
Oz and OA t OB, 00, so that 



then, with Euler s coordinate angles 6, </>, ^, 
= sin $cos = sin^sin 



-_p cos + sn ~ 
so that 



344 THE APPLICATIONS OF ELLIPTIC FUNCTIONS. 

G + CFr J__ G-CFr 1 
4 " 



__ 

2AF 1+C080 2AF l-cos0~ dt 
suppose ; and then 

, CFr-GcosO 



The equations given by Kirchhoff ( Vorlesungen iiber mathe- 
matische PhysiJc, p. 240) for a, fa y, the coordinates of with 
respect to fixed axes O a, /3, O y (O y parallel to Oz) are 



^- - ^-; .................. (9) 

at Vu &v 3w 

where a v 2 > a s denote the cosines of the angles between O a 
and OA t OB, 00; and fa, fa, fa, the cosines of the angles 
between O p and OA, OB, 00. 

Expressed b}^ Euler s coordinate angles, 

a^ cos 6 cos <p cos \^r sin <f> sin ^, 

2 = cos sin cos \/r cos (/> sin i/r, 

a 8 = sin cos ^r ; 

fa = cos cos sin i/r + sin cos \/r, 

fa = cos sin sin i/r + cos cos i/r, 

j# 3 = sin ^ sin i/r ; 
while p = s ^ n <f>0 sin cos \^, 

g = cos 00 + sin sin i/r, 

r = 0-}- cos \js ; 
so that, after reduction, 

Fa = A cos i/r ft + (Or J. cos ^)sin 6 sin i/r, 
^S = A sin \/r - (Or - A cos ^)sin 9 cos ^, 



Writing Fx for ^cos or Rw, equation (D) becomes 

n *i(x> if (IT rv ^V 2 - 1 
<& - l (a " Tj/ 1 4 2 

suppose, where n 2 T\D~pJ- 



APPENDIX. 345 

Denoting the roots of the quartic X = by x ot x v x 2 , x^ 
we may put, according to 151, 152, 



x-x = 





j. _ / - . ._ _ 3 

pu pc pc e B 
and now, when x oscillates between x. 2 and x%, 

u = nt + o) 3 . 

The letter u has been used here in two senses, to agree with 
the ordinary notation ; this need not however lead to confusion. 
Differentiating, 



(u f> 2 



2 <p ( u c) ^ 2c 



- c) + p(-u -f c) ; 

so that we must write v for 2c and u for u c, to agree with 
Halphen s notation. 
Now, to determine y, 






F = 



(F- 
^ 
nP 

so that, in a complete period 2^ of the motion, the point 
will have advanced parallel to O y a distance 
F 2 



also ( 152) 6^2c = coefficient of x 1 in X. 



346 THE APPLICATIONS OF ELLIPTIC FUNCTIONS. 

We now suppose that u = a makes #=1, and u = b makes 
x 1; then 



(pa - <pc)(<pu - pc) (p& - 

J?^Lf?l c _ _ .G CFr p 6 p c _ .6 + CFr 
(pa - pc) 2 ~ 4Jfy, " 

Then 

di\ls l _ \ p a(pit- 
</% (pa 



^ ! 

pa pc pM,-a 

= - !(( ~ c ) - KO + c) + fa 



and similarly 
; ^i=- 

and therefore 

ty = - .|Pu + i log 

where 



Also 

= -a; = 

$> 2 c($>u - pa)(p?> - pu) 
(pa - pc)(p6 - pc)(pw - p 



o-(a - c)a-(a + c )o-(6 - c)o-(6 + c)a- 2 (u - 
so that 



giving the projection on a plane perpendicular to 00 of the 
motion of a point on the axis OC t relatively to 0; also 



We find also, as in 224, that if the values a x and b l of u 
correspond to 



then 



APPENDIX. 347 

But now introduce the condition 

a + b = w (t , 
when, according to 282, \[s becomes pseudo-elliptic. 

Putting =tan- 






and, employing 6 instead of a, this may also be written 



so that 

1+X Q . l + X a l-X Q . l- 

and therefore each is equal to 1, and 



snce x + 

and, changing to the complementary angle. 

XR.X X-y 
X X Q .XXa 

-sin-i /^-^-*7_ M -i Ix-Vo-Xc^x 

V 2^2^"" V ~2-7 2 

with o. a > a;^ > a; > ic 7 > # . 
Differentiatin, 



so that 



-I ~~ iC" 



provided that n(x +x a ) = GjAF, n(l + x x a ) = Cr/A. 

The quartic ^ must therefore break up into the two- 

, ,. fe (7?- Gx Cr , 

quadratics ^-- + -1 and ^+-^--,--1$ Mid 



= (T&_W_( GX - CrF ^ 



so that the requisite relation when a + 6 = o>, is 



348 THE APPLICATIONS OF ELLIPTIC FUNCTIONS. 

Now 



so that sin 2 # sin 2= ^A r , sin 2 cos 2= 
and g=mt-\ls, 

where m = J (a; + a; a ) - J G/A F. 

Also, from (7) and (8), 

F(a cos r + 3 sin \/=A6 



= A n^/JT/sin 6 = An sin sin 2f ; 
JP( sin \/r /3 cos i/r) = (Cr xl cos 0\^)sin ^ 

CrF-GcozO 
-f^TO -An sm0 cos 2 

Therefore ^ a = A n sin 0(sin 2 cos \[s cos 2 sin \/r) 
= J.?i sin sin(2 \js) 
= Ans\n sin2m#- 
^/3 = ^1% sin Q c 
= ^4 sin 6 c 

Now in the motion of a point on 00, relative to 0, 
sin 0e^ = sin cos(mt g) + i sin 0sin(m 

_^r/ jX-X Q .Xa-X . lxp-X.X 

W" ~"2~ *V~ ^~T~ 

where a; = cos a 

When b a = wa t and V^i"^ or ^ s pseudo-elliptic, we 
shall find that 6r and Or are interchanged, and 



JCtJ 

then 2T-Cr z - ^ = ; ........................... (F) 

^L 

so that P*(V + v 2 ) = 



As a numerical exercise, we may take, in addition to (F), 

G = 4>AFn, Or =2^/7 An; 
then X = x 4 -Wx 2 +IG / J7x-l5 

= (a; 2 - 2J7x + S)(x* + 2 ^ - 5) ; 



= 60, (/ 8 = 88, ^ = 1+2^/3, 2 =-i>, 3 = 1-2^/3; 



APPENDIX. 34,9 

pa = - 8, pb = 1 ; a = a> 8 , 6 = Wl - Ja> 3 ( 225) ; 

pc = 2 v /7 + 3, p c = - 8^7 - 20, 9 2c = 5, p 2c = 
Now we shall find that 



=etc. 



sin 3 cos 3(nt - \fs} = ( - f + *J7 cos - cos 2 0)*, 
sin*08in3(nt ^) 

-2 cos + 7 cos/- cos 0- 



MISCELLANEOUS EXA^IPLES. 

1. Construct a Table exhibiting the connexion between th^ 
twelve elliptic functions 

sn u, ns u, dc u, cd it ; 
en u, ds u, nc u, sd u ; 
dn it, cs u, sc u, nd u. 

2. Construct a Table of the values of the sn, en, dn of 
u-\-mK-\-nK i in terms of sn u, en it, dn u ; also of the elliptic 
functions of ^(yiiiK-\-nK i), for m, tl=0, 1, 2, 

3. Prove that, accents denoting differentiation, 
(i.) sn it dn"it sn"u dn u = sn u dn u, etc. 

(suit) 2 , snt&sn tt, (sn u) 2 



(ii.) |(cnu) 2 , en it cn%, (cn tt) 2 



= /c /2 sn u cuudn u. 



(dnu) 2 , dnudn fi, (dn^t 

(G. B. Mathews.) 

4. Denoting by (m, n) the function 

sn (u m u n )cr\(u m + u n ) 



en u m 
prove that 

(4, 1)(4, 2)(4, 3X2, 3)(3, 1)(1, 2) + (4, 1)(2, 3) + (4, 2)(3, 

Denoting by A, B, G the functions 

z] sn( 7/)sn(s a;) 



z)sn(x y}" 
prove that ABC+A + B+C=0. 



350 THE APPLICATIONS OF ELLIPTIC FUNCTIONS. 

5. Prove that 

/~2u 

(i-) IK sn vdv = 2 tanh - l ( K sn 2 u). 

o 

(ii.) A sn(2 it -f a)du = tanh ~ x { K sn u sn(i& -f a) } . 

o 

fK 

(ill) / log us mfot = \TrK -\K log I/*. 
o 

6. Determine the orbit in which 

P = h?(u 3 +a 2 u 5 ), the apsidal distance being a. 

7. Rectify r*a*co8|0, 

8. Prove that the perimeter of the Cassinian Oval of 161 

. 4/3 2 # 

IB e^her - = 



and draw the corresponding curves. 

9. Prove that the length of the curve of intersection of two 
circular cylinders, of radius a and &, whose axes intersect at 

right angles, is 8a %l **^ K 2 = 2 /6 2 ; 





and verify the result when a = b. 

10. Prove that K and K satisfy the differential equation 

d ^n fr^l 1 7^ o 

d/c\ L(l ~ Lj dkr 4 

Deduce the relation 



die 
and thence deduce Legendre s relation ( 171). 

11. Prove that CTJ and C7 2 of 252 satisfy the differential 

2 - 
equation 



12. Deduce the Fourier series for snu, en u, dnit of 266, 
267 from the series for Zu of 268, making use of Landen s 
Transformations and of equations (28), (29), (30) of 264. 



MISCELLANEOUS EXAMPLES. 351 

13. Prove that 



n i s _ 

; 



.... , 



14. Prove that, if a variable straight line meets the curve 
Atf+Byt+Cx+D^Q 

, y 2 X^s y a ) then ( 166 ) 



.Vl 2/2 ?/3 

15. Denoting the integral 

/* xydx , ~ 
tf^x by fo) 
o 

where y is given as a function of # by the equation 
prove that, for three collinear points, 



16. Prove or verify that, with </ 2 = 0, the solution of Lame"s 
differential equation 



s = 



1 

s = 



(Halphen, Memoire sur la reduction des equations difftren- 
tidles, 1884.) 

17. Determine, by means of elliptic functions, the motion of 
liquid filling a rectangular box, due to component angular 
velocities about axes through the centre parallel to the edges. 

(Q. J. M., XV., p. 144 ; W. M. Hicks, FeZocify and 
Potentials betiveen parallel planes, p. 274.) 



352 THK APPLICATIONS OF ELLIPTIC FUNCTIONS. 

18. Prove that, with x = %7ru/w and A = ^/w ( 278), 



and thence convert the formulas (M) to (T) of 249 into- 
Jacobi s notation. 

19. Prove that ( 264, 20*) 

l/Q = T fl(l _ fr) = m 

r=l m 



20. Prove that 



(in.) ,_= II- 



21. Prove that, in Appendix II., p. 34G, 
O 2 



- b} - 



Work out the case of 



An* 



INDEX. 



PAGE 

Abel, - 4, 18, 145, 223, 277 

Abel s theorem, - 166, 249 

pseudo- elliptic integrals, - 2*28 
Abelian functions, - - - 175 
Addition theorem, 

of circular and hyperbolic 

functions,- - - - 112 

for elliptic functions, - 113, 117 

of second kind, 178, 180, 193, 226 

of third kind, - 193, 224 

algebraical form of, - - 142 

Theta function, - - - 192 
Allegret, - -, - 147 
Amplitude, - 278, 289 

hyperbolic, - - " - - 15 

of elliptic integral, - 4 
Anharmonic ratio, of four points, 53, 57 
Anomaly, mean, of a planet, - 14 
Apsidal angle, - - 260, 340 
Argand, - - - - - 46 
Argument, - 191 

Ballistic pendulum, Xavez, - 3, 12 

Bartholomew Price, 3 

Basset, . . 288 

Basset s Hydrodynamics, 219, 342 

Bertrand, - - - - 9 

Biermann, - - 77, 151 

Binet, - - - - 213 

Bjerknes, - . - 22 

Boys, ... . . 97 

Bravais, - - - - 340 

Brioschi, - - - - - 275 

G.E.F. i 



Burnside, W., - 
and Panton, 



PAGE 

- 38, 107, 172, 209 
148, 150 



Capacity, coefficients of electric, 287 
Capillary attraction, - - 89 
Cartesian ovals, - - 257, 262 

confocal orthogonal, - - 255 
Cassinian oval, 

area of, . 189 

rectification of, - - 164 
Catenaries, - - - - 76, 92 

of uniform strength, - - 92 
Catenoid, - - - - 95, 98 
Cauchy s residue, - - - 206 
Cayley, 

56, 62, 139, 142, 160, 280, 311, 327 
Central orbits, ... 76 
Chain, revolving, - - 67, 210 
Chasles, - - - 178 

Chevallier, - - - - 335 
Chrystal, .... 66, 277 
Circular functions, trigono 
metrical, - ... 6 
Clifford, - - - 17, 30, 284, 295 
Complementary modulus, - 9 

Conductivity, thermometric, - 285 
Confocal, ellipses and hyper 
bolas, - - - 184, 255 

quadric surfaces, - - 271 

paraboloids, - - - 273 
Cotes s spirals, 75, 190 

theorems, - - 289, 325 
Cubic substitution, 41 

353 



354 



THE APPLICATIONS OF ELLIPTIC FUNCTIONS. 



Cubicovariant, 
Cubin variant, - 
Cycloidal oscillation, 
Cycloids, - 



PAGE 

158 

62 

7 

190 



Darboux, - - 221, 236, 255 
Delambre, - 137 

De Moivre s theorem, - - 288 
De Sparre, - 233 

Despeyrous, - 236 

Descriminant, - - 44, 48, 158 

Discriminating cubic, - - 154 
Doubly periodic function, 

208, 254, 299 

Duplication formulas, - - 120 
Durege, - 117,227 

Dynamical problem, - - 74 

Elastica, - 87, 190 

tortuous, - - - 213 
Electrification of two insulated 

spheres, - - 287 

Electrode, - - - - 278 
Ellipse, rolling, - - - 71 

first negative pedal of, 73 
Elliptic functions, 

addition theorem, - - 142 

complex multiplication of, 

12, 203, 330 

double periodicity of, - - 254 

geometrical applications to 

spherical trigonometry, - 131 

multiplication of, - - 329 

reciprocal modulus, - - 24 

resolution of, into factors 

and series, - - 277 
Elliptic integral, 

of first kind, - - 4, 22, 30 

of second kind, - 64, 175, 209 

of third kind, 

108, 175, 191, 206, 302 

complete, - - 8 

definition of, 5 

degenerate, - - - 41, 57 

factor of third kind, - - 226 

general, - 200 

graphs of, - - - - 66 

half period of, - ... 13 



Elliptic integral coiitinued, 

inversion of, 30 

modulus complementary, - 9 

normalised, - - - 203 

quarter period of, - 8, 321 

quarter period, complemen 

tary, 9 

Tables of, - - 10, 11, 16, 177 

Weierstrass s defined, - - 42 
Enneper, - - - 61, 326 
Epitrochoid, - 190 
Eta function, - - 194,282 
Euler, - - - 142, 251 
Euler s addition equation, 144, 166 

constant, - - 281 

equations of motion, - 18, 101 

pendulum, - > 198 

Fagnano s theorems, - 182 

Forsyth, - - - 298 

Fourier, ... 66 

series, 285, 287 
Fricke, - 155 
Fundamenta Nova, - - - 310 
Fuss, 121 



Gauss, - - , - , 137, 322 
Gebbia, - - - - - 220 
Genocchi s theorem, - 264 
Geodesies, .... 95 
Glaisher, - 17, 33, 62, 116, 133, 194 
Governor, Watt s, - - 78 

Graphs of elliptic integrals, - 66 
Graves, ... - 178 

Gudermann, 5, 32, 90 

Gudermannian, - . 14 

Half period, imaginary, - - 44, 50 

- real, 43, 50 

Halphen, 128, 130, 206, 217, 276, 342 
Hammond, - - 256, 294 
Harmonic motion, 13 

Heat, conduction of, 284 
Helicoid, - - . - - 95 

Helix, ..... 20 
Hermite, 150, 158, 208, 215, 276, 335 
Herpolhode, - 101, 107, 207, 231 
algebraical, . . . 228 



INDEX. 



355 



62, 



Herpolhode, 

points of inflexion, 
Hess, 

Hessian, - 

Hicks, 

Hill, ... 

Hobson, - - 277, 

Holzmiiller, - 

Homogeneity, - - 203, 

Homogeneous variables, - 

Hooke s law, - 

Hoyer, - - - - 

Huygens, 

Hyperelliptic function, - 

integral, 
Hyperbolic amplitude, - 

functions, - 
Hypotrochoid, 



- 233 

- 233 
149, 156 
288, 351 

291 
280, 330 

- 257 
247, 270 

- 155 

94 

237 

6 

- 175 
168, 309 

15 
15 

- 190 



Icosahedron form, - - 156 

Imaginary period, - - - 254 
Induction, electric coefficient, 287 
Inflexion, points of, on herpol- 

hodes, - - - - 233 
Integrals, 

circular and hyperbolic, - 30 

hyperelliptic, - - 160 

poles of, - 45, 53 
Invariants, - - - 43, 62, 143 



absolute, - 

Jacobi, 

Jacobi s notation, 
Jenkins, - 
Jochmann, 



- 45, 49, 143 

- 5, 139, 160, 284 

- 18, 50 
84, 131 

- 278 



Kaleidoscope, - 293 

Kepler s problem, 14 

Kiepert, - 331 

Kirchoff, - - - 87, 344 
Kirchoff s kinetic analogue, - 214 

Kleiber, - 190 

Klein, - 35, 151, 271 

Kronecker, - ... 146 

Kummell, - 136 

Lagrange, - 131, 340 

Lambert s series, - - - 287 



Lame s differential equations 
210, 

parameters, 
Landen s point, - - 

transformation, - 55, 60, 

second transformation, 
Lecornu, 

Legendre, - 4, 18, 64, 
Legendre s relation, 
Lemniscate, ... 

rectification of, - - 



15, 



Linear substitution, 

transformations, - 

Lintearia, 

Lodge, 

Love, - - - 



216, 275 
272, 274 
23, 117 
186, 322 
120, 320 

- 212 
131, 323 
164, 178 

. 199 

- 33 

- 190 

- 143 
163, 316 

87 
278, 293 

- 293 



MacCullogh, - 
MacMahon, 
Mannheim, 
Maxwell, 
Mean anomaly, 
Mercator s chart, 

Sumner lines, 
Meridional part, 
Michell, - 
Minding, - 
Modular angle, 

equations, - 

equation of third order, 
Modulus of elliptic integral, - 
change f rom , and its reciprocal, 

complementary, - 

singular, 



179, 220 

- 147, 295 

- 221 
79, 89, 272, 287 

14 
17 
89 
17 

- 292 
. 121 

4 

- 323, 327 

323 
4 
24 
9 

331 



Morgan Jenkins, - 84, 131 

Motion, 

of a body in infinite liquid 

under no forces, - 219, 342 

of a projectile, resisting 

medium, 65 

of electricity or fluid, - - 266 

mean, of a planet, - 14 

Poinsot s geometrical repre 

sentation, .-- - 101 

solutions of Euler s equa 

tions of, - - 28, 101 
Miiller, 314 



356 



THE APPLICATIONS OF ELLIPTIC FUNCTIONS. 



PAGE 

Napier, - 137 

Nodoid, 95, 98 

Norm, - - - ....- - 278 

Octahedron form, - 157 

irrationality, - 317 
Orbits, central, . - . . 76 
Oscillations, 

cycloidal, 7 

quadrantal, - - 103 

rectilinear, - - - 25 

of pendulums, bell, etc., - 3 

vertical, of a carriage or ship, 82 

Parameter, 191, 207 

Pendulum, 1 

Euler s, - - 198 
- Navez, ballistic, - - 3, 12 

performing complete revolu 

tions, - 18 

period of, 8 

reaction of axis of suspension, 82 

simple equivalent, - - 3 

speed of, - 3 

spherical, 214 
Period, parallelogram, 46 

rectangle, - 270 
Poinsot, - - 233 
Poinsot s geometrical repre 
sentation of motion, - - 101 

Poles of integral, - - 255 

Polhode, - - 101 

separating, - 230 
Poristic polygons, Poncelet s, - 121 

heptagons, - 130 

pentagons, - - 128 

quadrilaterals, 126 

triangles, - 124 
Poundal, - 1 
Price, - 3, 79 
Pringsheim, 160 
Projectile, trajectory of, for 

cubic law of resistance, - 244 
Pseudo-elliptic, 242, 347 

integrals, Abel s, .- 228, 300 

Quadrantal oscillations, - - 103 

Quadri-quadric function, - - 148 



PAGE 

Quadric surfaces, confocal, - 271 

transformations, - - 35, 321 
Quadrinvariant, 62 

Radian, - 1 

Real half period, - 43 

Reciprocant, - ... 294 
Reduction, formula of, - - 63 
Reversion of series, - - - 202 
Revolving chain, - - 67, 210 
Richelot, - 121, 164 

Riemann, 312 

differential relation, - - 316 
Roberts, - 162, 165, 199 
Robertson Smith, - 291 
Rocking stone, - .- - 198 
Rolling and sliding cone, 108 
Routh, - 3, 28, 101, 217, 238 
Russel, - 62, 149, 151, 158, 327 

Salmon, - 149, 157, 162, 178, 222 
Schwarz, - - 26, 46, 157, 298 

Sextic covariant, 150, 157, 163, 321 
Siacci, - - 220 

Sigma function, - . - 201 
Simple harmonic motion, - 13 
Simpson, - - - - . - 9 
Slade, - - .. .. 233 

Smith, - 27, 222 

Spherical pendulum, - - 214 
Spherical trigonometry, - - 169 

geometrical application of 

elliptic functions to, - 131 

Spinning top, - - - . - 214 

Spiral, Cotes s, 75 

Steiner, - 121 

Substitution, linear, 143 
Sumner lines on Mercators 

chart, 89 

Surface, special minimum, - 26 
Swinging body, internal stresses 

of, - 84 

Sylvester, 221, 294 

Syzygy, - 150, 156 

Tables of elliptic functions, 

10, 11, 16, 177 
Tait, - - - .- .. .;-. 87 



INDEX. 



357 



Talbot s curve, 
Temperature, stationary, 
Theta function, 

addition theorem for trans 

formation of, 
Thomson, Sir W. , - 
Thomson, J. J., 
Top, spinning, - 

degenerate cases of, 
Tortuous elastica, - 
Trajectory of projectile for 

cubic law of resistance, 2- 
Transformation of elliptic func 
tions, ... 

physical application of, 

first real, 

second real, 

theta functions, - 

third and higher orders, 
Triplication formulas, 
Trochoids, 

Turning points, 



PAGE 




PAGE 


73 


Unduloid, - . . . 95, 98 


266 






192, 282 


Vector function, 


278 


ins- 


Vortex, - 


291 


- 325 






3, 86, 287 


Wallis s theorems, - 9, 63, 


176 


119, 145 


Wantzel, - 


213 


214 


Watt s governor, 


78 


- 241 


Weber - 


3OA 


213, 237 


Weierstrass, - - - 151, 


OOU 

296 


for Weierstrass s elliptic functions, 


42 


244, 316 


double periodicity of, - 


264 


nc- 


notation, .... 


146 


- 305 


Wilkinson, - 


116 


305 


Woolsey Johnson, - - 116, 


133 


306 


Wright, - 


17 


306 


Wolstenholme, 


130 


- 325 






323 


Zeta function, - - - 191, 


201 


120 


addition equation of, - 


204 


190 


Zimmermann, - 


293 


59 


Zonal harmonics, 


311 



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